Marianna Belotti
email: marianna.belotti@caissedesdepots.fr

Stefano Moretti
email: stefano.moretti@dauphine.fr

Maria Potop-Butucaru
email: maria.potop-butucaru@lip6.fr

Stefano Secci
email: stefano.secci@cnam.fr

Game Theoretical Analysis of Cross-Chain Swaps

In this paper we address the distributed cross-chain swap problem in the blockchain context where multiple agents exchange assets across multiple blockchain systems (e.g. trading Bitcoins for Litecoins or Ethers). We present a mathematical framework allowing to characterize blockchain swap protocols as the combination of a publishing and a commitment phase, where contracts are respectively published and then committed. We characterize the equilibria of existing cross-chain swap protocols (i.e., blockchain swap protocols exchanging assets among different blockchains). More precisely, we prove that following a swap protocol characterized by concurrent publishing of exchange contracts and snap (immediate) assets transfers is a Nash equilibrium. Furthermore, we prove that for protocols with a sequential publishing and commitment of the assets transfers, following the prescribed protocol is a sub-game perfect equilibrium.

I. INTRODUCTION

The modern economy is moving to a new era where economical transactions use cryptocurrencies instead of fiat money. Crypto-currencies (i.e., Bitcoin, Ethereum, etc.) are based on the use of blockchains, which are basically transactional systems governed by decentralized protocols. Blockchains pave the way for a new approach to organize and sustainably maintain long-term transactions as well as high-level services. It is interesting to note that the number of blockchains that currently hold the head of newspapers has gone from one in 2008 (the famous Bitcoin blockchain [START_REF] Nakamoto | Bitcoin: A peer-to-peer electronic cash system[END_REF]) to a few tens in 2018 such as Hyperledger [START_REF]Hyperledger Architecture Vol.1, Introduction to Hyperledger Business Blockchain Design Philosophy and Consensus[END_REF], Ethereum [START_REF] Buterin | Ethereum White-paper[END_REF], Zcash [START_REF] Green | Bolt: Anonymous payment channels for decentralized currencies[END_REF], Corda [START_REF] Hearn | Corda: A distributed ledger[END_REF], Ripple [START_REF] Schwartz | The Ripple protocol consensus algorithm -White Paper[END_REF], Tendermint [START_REF]Tendermint Byzantine-fault tolerant state machine replication[END_REF] etc. Each of these systems has its own modus operandi, its own governance and even its own way of agreeing on a common history. Each system has its own advantages that make it attractive for various applications and geopolitical contexts. We are witnessing the creation of several ecosystems, each with its own currency and governance.

Similar to modern international economical exchanges which are based on different government-issued currencies, inter-blockchain exchanges must be based on common rules resilient to attacks, failures or malicious behaviors affecting the network. There are currently several operational systems for achieving interoperability between different blockchains such as Kybernetwork [START_REF]Kyber: An On-Chain Liquidity Protocol[END_REF], Aion [START_REF]Open Application Network[END_REF], Cosmos [START_REF]Cosmos: A Network of Distributed Ledgers[END_REF] or Polkadot [START_REF]Polkadot: Vision for a Heterogeneous Multi-Chain Framework[END_REF]. These systems can be classified into two categories according to their decentralization level: systems that use a trusted thirdparty to validate transactions or systems that realize it directly between blockchains without the need of a trusted third-party. In order to execute an exchange or a swap (i.e., a set of transactions between parties), transacting agents (i.e., blockchain users) are provided with a protocol to stick to. A protocol in this case consists of a specific sequence of instructions agents should perform to preserve the ACID properties [START_REF] Lewis | Databases and transaction processing: an application-oriented approach[END_REF] of the individual transactions or exchanges; that is, Atomicity: the all-or-nothing occurs and each participant must know which state he or she is in; Consistency: each successful transaction by definition commits only valid results; Isolation: transactions run independently; Durability: transactions cannot be abrogated after commitment.

Atomic cross-chain swaps fall into the class of the so called TAST (i.e., Token Atomic Swap Technology) [START_REF] Borkowski | Towards atomic cross-chain token transfers: State of the art and open questions within tast[END_REF] research ideas aiming at making blockchains interoperable. Differently from decentralized exchanges -initiatives recently emerged to remove the need of trust on traditional exchange services (e.g., the format proposed by the Ethereum DEX protocols [14; 15] atomically swapping ERC-20 tokens on the Ethereum blockchain) -atomic swaps have not limits in operating crosschain. The very first atomic swap solution has been proposed for Bitcoin by Nolan [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] making use of hash-time locked contracts enabling conditional assets transfers. Nowadays few platforms actually support cross-chain exchanges that at this stage are still slow and inefficient. Decred [START_REF]Decred cross-chain atomic swapping[END_REF] implements Nolan's logic on UTXO-based premissionless blockchains such as Bitcoin Cash [18; 19; 20], Litecoin [START_REF]Litecoin core integration/staging tree[END_REF], Qtum [START_REF]Qtum project[END_REF], etc. BartherDEX [START_REF]Komodo barterdex[END_REF], part of the Komodo project [START_REF]Komodo (advanced blockchain technology, focused on freedom)[END_REF], represents a cross-chain solution that matches orders and defines the swap protocol. Blockchain.io [START_REF]Blockchain.io (your gateway to the internet of value)[END_REF] implements atomic crosschain swaps by combining centralized components (order matching) with decentralized ones (trade settlement and execution). Therefore, research now focuses on hybrid swap protocols, replacing decentralized commitment/locking schemes (hash-locks) with centralized ones, resulting more attractive and efficient. AC3TW and AC3WN [START_REF] Zakhary | Atomic commitment across blockchains[END_REF] protocols propose atomic cross-chain swaps respectively with centralized and "decentralized" trusted authorities (i.e., an external agent and an external blockchain) acting as witnesses. According to Arwen protocol [START_REF] Heilman | The arwen trading protocols[END_REF] crypto-assets are swapped through centralized exchange services that in no way acquire the assets custody. XClaim [START_REF] Zamyatin | Xclaim: Trustless, interoperable, cryptocurrency-backed assets[END_REF] overcomes blockchain data-structures incompatibility by swapping cryptocurrency-backed assets.

It should be noted that different swap protocols differ essentially in the involved parties. The set of swap participants can be composed only of the asset owners (e.g., as in [START_REF] Herlihy | Atomic cross-chain swaps[END_REF]) or by owners accompanied by a trusted third party (e.g., as in the AC3TW protocol [START_REF] Zakhary | Atomic commitment across blockchains[END_REF]).

To the best of our knowledge, there is (i) no formal analysis on the structure and the properties blockchain swap protocols satisfy and, (ii) no game theoretical modeling of participants strategic interactions (i.e., to follow or not to follow the prescribed protocol). More precisely, Nolan [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] presents the Bitcoin swap protocol in a functional manner. In [START_REF] Herlihy | Atomic cross-chain swaps[END_REF] the author provides a partial game theoretical analysis (specific to the protocol) presenting no structural characterization of the possible equilibria of the system (see Section III-A).

Authors in [START_REF] Zakhary | Atomic commitment across blockchains[END_REF] analyze the atomicity violations characterizing the protocol in [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF]. However, no game theoretical result is provided.

According to recent studies [30; 31] proposed swap protocols are not properly analyzed neither from the structural point of view, responsible for their atomicity, nor from the strategical point of view making them satisfying liveness and safety properties (i.e., the protocol terminates in a valid state). More precisely, it is important to analyze the behavior of the swap participants that are rational agents actively participating in the exchange by following or not the prescribed protocol according to their own objective function.

A. Our contribution

We propose a generic game theoretical framework that formalizes the swap problem and characterizes blockchain swap protocols by clearly separating the contracts publishing phase and their commitment phase. Furthermore, we prove that (i) following a swap protocol characterized by an effective decision function (when players have the power to accept or decline the desired assets) is a subgame perfect equilibrium in dominant strategies and, that (ii) following a swap protocol characterized by concurrent publishing and snap (immediate) commitment is a Nash equilibrium. Our generic framework allows us to characterize equilibria of two representative recent protocols presented in [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] and [START_REF] Zakhary | Atomic commitment across blockchains[END_REF] respectively. In the case of the protocol proposed in [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] and generalised in [START_REF] Herlihy | Atomic cross-chain swaps[END_REF], following the protocol is the unique subgame perfect equilibrium (in dominant strategies), while in the case of the protocol proposed in [START_REF] Zakhary | Atomic commitment across blockchains[END_REF], following the protocol is a Nash equilibrium.

II. SWAP GAMES

In this section we propose a formal definition of the swap problem and a formalization of the corresponding blockchain swap protocol that can be atomic or not. The latter consists of two different phases (i.e., publishing and commitment of transfers) and represents a particular strategy of a swap game that players can choose to adopt or not.

A. Swap problem and swap protocol

In a general swap problem, swapping parties aim at exchanging assets among themselves. A swap protocol defines the set of asset transfers and the order in which they should be executed. Given a set of assets and the corresponding owners, a swap consists of an asset ownership exchange within the set of owners.

Definition 1 (swap problem).

A swap problem is defined as a tuple A, O, b 0 , b * , (u i ) i∈O where:

• A = {1, . . . , m} is the set of assets to be swapped;

• O = {1, . . . , n} is the set of owners or agents participating in the exchange. We consider in the following that n ≤ m since each owner owns at least one asset; • b 0 , b * : A → O are the original and the desired ownership maps -both surjective -such that ∀a ∈ A, b 0 (a) = b * (a) and;

• u i is the payoff function for owner i ∈ O over bundles of assets in 2 A such that u i (b -1 0 (i)) < u i (b -1 * (i)) (i.e.
, each owner i strictly prefers the desired bundle of assets to her original bundle) and ∀S, T ∈ 2 A : S ⊆ T , u i (T ) ≥ u i (S) (i.e., a larger bundle is strictly preferred to a smaller one), for each i ∈ O.

The representation with two different surjective functions b 0 , b * describes swaps as ownership exchanges problems. Agents participate in a swap with the asset(s) they aim to exchange for others, preferring the desired new asset(s) to finding themselves as in the initial configuration. Moreover, we assume owners' payoff function increases monotonically with the size of the asset bundle.

The transition from an initial configuration to a post-swap configuration is defined by the corresponding swap protocol consisting in a sequence of operations to be executed in a certain order. In centralized swap protocols a central role in the swap is played by a trusted third party. That is, asset ownership is transferred first to the trusted third party that in turn transfers assets back to the new owners. On the other hand, decentralized swap protocols contemplate ownership transfers within asset owners only; the latter agrees on a particular swap configuration (i.e., the assets to exchange) without trusting each other.

In order to formally define a swap protocol we need to introduce first, the structure of a decentralized exchange protocol, consisting of a sequence of asset(s) transfers to be committed.

Definition 2 (decentralized exchange protocol). Let σ = {(A k , O k , X k ) : |A k | ≥ |O k |} k , k ∈ {1, . . . , t}, t ∈ N : t ≤ m
be a sequence of exchanges where,

• A k ⊆ A specifies the subset of assets involved in the exchange at step k; • O k ⊆ O specifies the subset of owners involved in the exchange at step k;

• X k : A k → O k (surjective) specifies the owner X k (a) ∈
O k of any asset a ∈ A K at step k; A sequence σ defines a decentralized exchange protocol that engenders a sequence of maps b σ 1 , b σ 2 , . . . , b σ t : A → O such that for all k ∈ {1, 2, . . . , t}:

• b σ k (z) = b σ k-1 (z), ∀z ∈ A \ A k ; • b σ k (z) = X k (z), ∀z ∈ A k , where we set b σ 0 = b 0 .
So, the triple One may consider a sequence of exchanges involving first owners 1 and 2, then 1 and 3 and finally 2 and 3. Precisely,

(A k , O k , X k ) specifies that the asset set A k ⊆ A is transferred at step k to owners O k ⊆ O according to X k . Note that
σ = ({a, c}, {1, 2}, {X 1 (a) = 2, X 1 (c) = 1}), ({b, e}, {1, 3}, {X 2 (b) = 3, X 2 (e) = 1}), ({d}, {2}, {X 3 (d) = 2}).
Sequence σ engenders a sequence of ownership maps such that b

0 = (1, 1, 2, 3, 3), b 1 = (2, 1, 1, 3, 3), b 2 = (2, 3, 1, 3, 1), b 3 = (2, 3, 1, 2, 1) = b * .
The sequence σ defines, for each protocol step k, (i) the assets A k whose property is to be transferred, (ii) the new assets' owners O k and, (iii) the function X k assigning the precise owner to each asset. The protocol is decentralized as ownership transfers take place among the owners themselves. At each step k an asset can change owner, the final configuration at time t provides the final asset owner.

In the simple case where agents agree on exchanging assets through a single ownership transfer at each step k, the corresponding single-swap protocol is represented by the sequence

σ = {(a k , o k ) : o k ∈ O, a k ∈ A} k∈{1,...,t} , t ∈ N : t ≤ m.
Protocol σ describes a general asset exchange agreement, and not necessarily a swap where the transfer of an asset ownership exists only if associated with the transfer of another one. More precisely, swap participants are interested in the final (at time t) and not temporary (intermediate at step k = t) acquisition of one or more assets owned by other agents. In a swap protocol, each asset changes owner only once. Definition 3 (decentralized swap protocol). A decentralized swap protocol is defined as a decentralized exchange protocol where the set {A k : k = 1, . . . , t, t ∈ N : t ≤ m} is a partition of the asset set A.

Example 1 provides a decentralized swap protocol σ since {{a, c}, {b, e}, {d}} is a partition of A.

From the formalization introduced in Definition 2 it is possible to derive a graphic representation, by means of a digraph D = (V, E), as the one proposed by Herlihy in

B. Atomic swap protocol

Whenever swapping parties do not trust each other it is in their interest to ensure that no participant can take advantage from the swap agreed on. The protocol must be constructed in such a way that the swap is performed in its entirety or no asset transfer is committed (i.e., all-or-nothing). In the case of failures during the protocol execution, every swap participant must be able to regain possession of the original owned assets.

Definition 4 (efficient decentralized swap protocol

). A decen- tralized swap protocol σ is said to be efficient if the engendered sequence b σ 1 , b σ 2 , . . . , b σ t is such that b σ t = b * . Definition 5 (atomic decentralized swap protocol). A decen- tralized swap protocol σ is atomic whenever it is efficient (in the sense of Definition 4) or b σ t = b 0 .
In order to prevent participants to externally exchange the assets involved in the swap, the protocol requires assets to be locked in specific transactions (i.e., they cannot be the object of other transfers). Once locked, the transfer commitment allows every participant to redeem the new swapped asset(s). Moreover, due to the atomicity requirement:

(i) any commitment should be conditioned on the correct asset locking i.e., transfers are committed only when all assets of the swap are correctly locked; (ii) consequently to failures in the assets locking, the initial situation must be restored; (iii) once an asset transfer is committed all the other transfers have to be committed, too.

When considering blockchains -decentralized trustless environments -swap protocols need to be atomic according to Definition 5. Conditional transfers in blockchain systems are implemented with distributed contracts i.e., scripts executed on blockchain nodes that can enforce and regulate relationships among network actors. A widely adopted commitment scheme is the crypto-primitive hash-lock [START_REF] Wiki | Hashlock[END_REF] i.e., asset ownership is locked with a hash value h that is the outcome of a one-way function H with a secret s as input (i.e., h = H(s)) and can be unlocked only when s is revealed. Transfers are conditioned on the hash constraint guaranteeing that assets to be swapped have been properly locked. Whenever transfers are not committed, due to atomicity, the initial ownership configuration needs to be restored; this can be implemented with time-locks [START_REF]Timelock[END_REF] i.e., a contract primitive restricting the asset transfer until a specified future time. Hashlocks and time-locks properly combined enable conditional asset transfers necessary to satisfy requirements (i) and (ii) of a swap protocol. Hence, when considering a blockchain swap, an asset transfer consists of an atomic distributed contract that, according to certain protocols [16; 29], coincides with a distributed hash-locked contract (HTLC).

The latter needs to be correctly implemented, published and validated on the network (i.e., part of the valid transaction history) before being committed. Concerning requirement (iii), a mechanism that forces committing all transfers is needed in order to avoid possible atomicity violations (see Section III-A3). However, having a forcing scheme comes at a price, i.e., loss of decentralization (see Section III-B).

C. Atomic blockchain swap protocol

In this work we consider blockchain swap protocols characterized by two distinct phases: a first phase in which transfers (i.e., hash-locked contracts) are published and a second one where they are committed. To achieve atomicity, the order in which operations in the two phases are executed matters (see Section III). The commitment phase has to be conditioned by the execution of the publishing phase; all the asset transfers have to be published before being committed. Therefore, a blockchain swap protocol is characterized by a publishing protocol σ P followed by a commitment protocol σ T . While the first protocol (σ P ) is a simple sequence of transfers, the second one (σ T ) is a decentralized swap protocol according to Definition 3.

In order to analyze the participants' strategic behaviors the protocol needs to specify the schedules in which agents perform operations. Therefore, given a sequence of operation σ, a set of asset owners O and a centralized trusted authority τ , we define a decision function F : {1, . . . , t} → O ∪ {τ } as a map that specifies the agent(s) F (k) responsible for the publication (or commitment) of the transfer (A k , O k ). Note that the agent(s) called to decide on the transfer can be either asset owners (i.e., F (k) ∈ O) or an external trusted actor (i.e., F (k) = τ ). Definition 6 (decentralized blockchain swap protocol). A decentralized blockchain swap protocol or simply a blockchain swap protocol is defined by the pair (σ P , σ T ) where

• σ P = {(A j , O j )} j∈{1,...,t P } , t P ∈ N : t P ≤ m, A j ⊆ A and O j ⊆ O is a sequence such that ∀j ∈ {1, . . . , t P }, O j = {o ∈ O : o ∈ b * (A j ) ∨ o ∈ b 0 (A j )} and, • σ T = {(A k , O k , X k )} k∈{1,...,t T } is a swap protocol engendering the sequence of maps b σ T 1 , . . . , b σ T t T : A → O according to Definition 3.
We associate to each sequence σ P , σ T the corresponding decision function F P , F T defined above.

Let us note that the publishing sequence σ P is constructed in such a way that transfers that can be published are of type (A j , b * (A j )) or (A j , b 0 (A j )). That is, blockchain transactions can transfer the asset ownership to desired owners and original owners only.

Definition 7 (atomic blockchain swap protocol). An atomic blockchain swap protocol consists of a pair (σ P , σ T ) where the engendered ownership map at time t of the commitment protocol coincides with the desired one; b σ T t T = b * or with initial one; b σ T t T = b 0 . 1) Phases separation: We have defined a blockchain swap protocol as a publishing sequence followed by a commitment emphasizing the precedence of the first phase over the second. However, it is necessary to condition the execution of the commitment protocol to the publication of all the contracts in order to have an atomic blockchain swap protocol. The following definition formalizes such a commitment requirement.

Definition 8 (commitment requirement). Given a blockchain swap protocol (σ P , σ T ) whenever there exists an asset transfer (using the swap structure previously given) that is not correctly published, then no asset transfer is committed. Formally, if, in σ P , ∃ j ∈ {1, . . . , t P } :

O j ∩ b 0 (A j ) = ∅ then, in σ T , b σ T k = b 0 ∀k ∈ {1, . . . , t T }.
Every blockchain swap protocol has to meet the commitment requirement defined above in order to be atomic. Hence, Definition 8 is a necessary condition (but not sufficient, see Section III-A3) for atomicity. Indeed, whenever an asset transfer is not correctly published the time-lock acts by not modifying the original asset ownership, i.e., transferring the asset back to the original owner (see [34; 35] for more details).

Focusing separately on the commitment protocol, it is possible to derive initial properties on blockchain swaps.

2) Commitment protocol: As stated in the following proposition, whenever an asset transfer is not committed, the swap participant supposed to acquire the asset(s) ends up with less assets than expected while the original asset owner finds himself with an extra asset. Next proposition does not depend on the exact specification maps X k , so we omit them in the sequence σ T for the sake of simplicity.

Proposition 1. Given a commitment sequence σ T = {(A k , O k , X k )} k∈{1,...,t T } , t T ∈ N : t T ≤ m then, replacing O k by b σ T k-1 (A k ) in σ T i.e., considering a new sequence σ k T = (A 1 , O 1 ), . . . , (A k-1 , O k-1 ), (A k , b σ T k-1 (A k )), (A k+1 , O k+1 ), . . . , (A t T , O t T ), for some k ∈ {1, . . . , t T }, implies that: (i) b σ k T t T -1 (O k ) ⊆ b σ T t T -1 (O k ) and, (ii) b σ k T t T -1 (b σ T k-1 (A k )) ⊇ b σ T t T -1 (b σ T k-1 (A k )). Proof. The claims follow from the fact that A k is not given to O k in σ k
T but it is given back to the original owners.

At time t, assets A k do not necessarily belong to O k thus, O k could find themselves with no asset i.e., b

σ k T t -1 (O k ) = ∅.
Moreover, the original asset owners find themselves with both the original assets and the swapped ones.

Let us denote by σ K T , with K ⊆ {1, . . . , t T } the sequence of pairs obtained by replacing

(A k , O k ) in σ T by (A k , b σ T k-1 (A k
)) for any k ∈ K (i.e. by the owner(s) of A k at step k -1).

3) Sequential phases and beyond: The formalization provided by Definition 6, enables capturing:

1-Single-asset and multi-assets swaps depending on the cardinality of the asset set A k at step k in the commitment protocol. 2-Swap protocols with sequential publishing and commitment, both single-asset and multiple-assets, where ownership transfers are published and committed according to a precise temporal order. Every crypto-asset transfer has to be executed before or after another one. In Section III-A we show that the sequentiality of the publishing phase combined with the leader role of a swap participant guarantees that the commitment ownership is verified. 3-Swap protocols with concurrent publishing where transfers are no longer published according to a given time order, but may be concurrently created and propagated to the blockchain network. In fact, it is not essential that the various transfers are published one at a time but only that there is a clear distinction between the publication and the commitment phase. Therefore, the transfers publication can take place in a concurrential way and this is captured by the sequence σ P where at step j : j ∈ {1, . . . , t P } multiple assets transfers (A j , O j ) are published. 4-Swap protocols with snap commitment, where assets transfers are all committed in the same time.

Proposition 2. A blockchain swap protocol with a snap commitment scheme satisfying the commitment requirement as in Definition 8 is atomic.

Proof. By contradiction, if b σ T t T = b 0 ∧ b σ T t T = b * then, since no problem in the publishing occurs, there should be a situation where some assets transfers are triggered and some others are not. However, this contradicts the snap commitment.

4) Decision function:

Let us focus on the decision function F previously defined. A blockchain swap can be completely driven by the asset owners only, in that case the function outcomes are all elements of O. On the other hand, we can have the intervention of an external actor τ entrusted for committing assets transfers. More precisely, the external trusted actor cannot publish blockchain contracts in behalf of asset owners since blockchain swap protocols contemplate ownership transfers among owners only (as in Definiton 6). Considering the publishing phase we can notice that every asset owner is in charge of publishing the signed contract transferring the asset ownership. In case of multi-assets transfers a multisig scheme [START_REF] Harn | Digital multisignature with distinguished signing authorities[END_REF] can be adopted.

Definition 9 (ownership requirement). In a blockchain swap protocol, the decision function F P corresponding to the publishing sequence σ P is such that F P (j) = b 0 (A j ) ⊆ O for any j ∈ {1, . . . , t P }, t P ∈ N : t P ≤ m.

Focusing on the commitment phase we can derive the following definition.

Definition 10. A decision function F T is effective on σ T if and only if F T (k) = O k for any k ∈ {1, . . . , t T }, t T ∈ N : t T ≤ m.
Whenever the decision function is effective, agents in O k have the power to accept or decline (i.e., redeem or not) the acquisition of the asset(s) in A k . Now, we are ready to analyze the strategic behaviors of the swap participants in a blockchain swap protocol. In this paragraph we associate a specific blockchain swap protocol (σ P , σ T ) (e.g., sequential, concurrent publishing and snap commitment) with the corresponding game (in strategic or extensive form).

D. Swap problem as a game

In the economic sphere, the strategies adopted by the agents mostly concern the maximization (or minimization) of their outcomes (e.g., prices or costs). Differently, in decentralized distributed systems (e.g., blockchains) strategies are related more on following or deviating from a prescribed protocol. In the blockchain environment several network and consensus algorithm attacks have been analyzed through game theoretical models [37; 38; 39] in order to qualify systems' strategic robustness. Considering decentralized swap protocols, the participants, intervening in both commitment and publishing phase, are more or less incentivized to stick or not to the protocol regulating the swap. The outcomes of a single agent, varying according to personal strategies, result depending from other agents strategic behaviors, too. What is common knowledge among the swap participants is the swap protocol structure that is represented by the pair (σ P , σ T ).

1) Preliminary notions on games:

A strategic form game is a tuple N, (S i ) i∈N , (u i ) i∈N where N = {1, . . . , n} is the set of players, S i is a finite set of pure strategies or actions for player i and u i : j∈N S i → R is a payoff function specifying for each strategy profile or state s = (s i ) i∈N ∈ j∈N S i player i' s payoff u i (s) ∈ R, for each i ∈ N .

Given a strategy profile s = (s i ) i∈N ∈ j∈N S i , in the following, s -i will denote s from which the strategy of player i is removed, i.e. s -i = (s 1 , . . . , s i-1 , s i+1 , s n ) and (x, s -i ) will denote the strategy profile s from which s i is replaced by x ∈ S i , i.e. (x, s -i ) = (s 1 , . . . , s i-1 , x, s i+1 , s n ). We say that x is a best response to s -i when u i (x, s -i ) = max y∈Si u i (y, s -i ). A state s = (s i ) i∈N ∈ j∈N S i is a (pure) Nash equilibrium of the strategic game N, (S i ) i∈N , (u i ) i∈N , if for every player i ∈ N , it holds that s i is a best response to s -i for each i ∈ N .

An extensive form game (with perfect information) on N is defined as a tuple N, T, P, (A h ) h∈V , (u i ) i∈N where T = (V, E) is a tree with root v 0 and the set of end nodes denoted by Z; P : V \ Z → N is a function assigning to each non-end node (also called decision node) a player in N ; A h = {(x h , x i ) ∈ E} for each node h ∈ V \ Z is the set of edges going from node h to some other node and represents the set of actions at node h; u i : Z → R is a payoff function for player i ∈ N . A pure strategy s i for player i ∈ N is a map assigning an action (edge) a ∈ A h to every node in P -1 (i) and we denote by S i the set of all pure strategy. Any strategy profile (s 1 , . . . , s n ) results in an end node in Z. So, to any extensive form game N, T, P, (A h ) h∈V , (u i ) i∈N we can associate a strategic form game N, (S i ) i∈N , (u i ) i∈N . A Nash equilibrium of an extensive form game is a Nash equilibrium of the associated strategic form game. A subgame of an extensive form game (with perfect information) on node v ∈ V \ Z is another extensive form game (with perfect information) obtained as a part of the directed tree starting at the decision node v. A subgame perfect equilibrium is a strategy profile that induces a Nash equilibrium on any subgame. For more details see, for instance, the book [START_REF] Peters | Game theory: A Multi-leveled approach[END_REF].

2) Blockchain swaps as a game: As discussed in Section II-C, blockchain swap protocols in both single and multiassets case, can be characterized by sequential publishing and commitment, concurrent publishing and snap commitment. For those protocols characterized by concurrent moves (i.e., concurrent publishing) we can adopt a representation with general strategic games. Definition 11. Let A, O, b 0 , b * , (u i ) i∈O be a swap problem, let (σ P , σ T ) be a blockchain swap protocol and let F P : {1, . . . , t P } → O and F T : {1, . . . , t T } → O ∪ {τ } be decision functions. We associate the strategic game form N, (S i ) i∈N , (u i ) i∈N such that:

• N = O is the set of players; • (S i ) i∈N = {action 1, action 0} is the set of pure strategies for player i consisting in the pair (follow the protocol, not follow the protocol) labelled respectively as action 1 and action 0; • (u i ) i∈N : u i : j∈N S i → R is the payoff function for asset owners N = O evaluating the outcomes of type b σ T t T that is, for every player i we have u i (b σ T t T ) -1 (i) . We model protocols with sequential phases with extensive form games in order to represent the sequencing of swap participants' moves and the fact that at each decision point asset owners know the moves history so far. Let us recall that blockchains involved in a swap are of public nature with open read access. Definition 12. Let A, O, b 0 , b * , (u i ) i∈O be a swap problem, let (σ P , σ T ) be a blockchain swap protocol and a let F P : {1, . . . , t P } → O, F T : {1, . . . , t T } → O ∪ {τ } be decision functions. We associate the extensive game form Γ σ = N, T, P, (A h ) h∈V , (u i ) i∈N such that:

• N = O is the set of players;

• T is a (binary) directed tree such that each directed path from the root v 0 ∈ V to an end node v ∈ Z is formed by precisely t + 1 nodes and t arcs (V is the set of the nodes of the tree T and Z ⊂ V is the set of leaf nodes); • P (v) = F P,T (l(v)), for each v ∈ V \ Z, is the publisher/activator of the asset transfer(s) at step l(v) in the protocol (σ P , σ T ), where l(v) is the number of arcs between v 0 and v on the unique path from v 0 to v (we assume that l(v 0 ) = 0); • A h for all h ∈ V is formed by two outgoing arcs in h; one arc in A h is labeled with action 1 (i.e., follow the protocol) and the other one, action 0 (i.e., not follow the protocol) for any h ∈ V \Z. So, a unique path from v 0 to an end node z ∈ Z identifies a binary vector p z ∈ {0, 1} t such that p z k is the label of the arc starting from node v with l(v) = k on the path from v 0 to z.

• Any end node z ∈ Z is associated with a unique outcome corresponding to the map b σ K T t T where K ⊆ {1, . . . , t T } is such that p z k = 0 for any k ∈ K, and p z k = 1 for any {1, . . . , t T } \ K. So, for any i ∈ O, the outcome b T,P,(A h ) h∈V , (u i ) i∈N be the extensive form game associated with the swap problem A, O, b 0 , b * , (u i ) i∈O and the blockchain swap protocol with sequential phases (σ P , σ T ) and let F T : {1, . . . , t T } → O ∪ {τ } be a decision function. If F T is effective on σ T , then the strategy profile (ŝ 1 , . . . , ŝn ) that specifies action 1 (follow the protocol) at any node is the unique subgame perfect equilibrium (in dominant strategies).

σ K T t is evaluated by i with the payoff function u i (b σ K T t ) -1 (i) . Proposition 3. Let Γ σ = N,
Proof. For each node v, by the fact that F T is effective, we have that v) . So, at each decision node v ∈ V \ Z, if player P (v) specifies action 0 (not follow the protocol) at node v, then by the first claim of Proposition 1, player P (v) ends up with a set of assets that is contained in the one that player P (v) would obtain if she/he specifies action 1 at node v. So, the utility of player P (v) is larger if it chooses, at each decision node, action 1 than action 0. It follows that at each node v action 1 strictly dominates action 0 for player P (v).

P (v) = F T (l(v)) = O l(
Notice that if (σ P , σ T ) is efficient, than in Proposition 3 the unique subgame perfect equilibrium (in dominant strategies) corresponds to the desired outcome b * . Proposition 3 shows that, whenever players in the game have to decide whether to accept or decline an asset acquisition, they are incentivized to follow the protocol by accepting the desired asset. However, a stronger result can be proved. Proposition 4. Let Γ σ = N, T, P, (A h ) h∈V , (u i ) i∈N be the extensive form game associated with the swap problem A, O, b 0 , b * , (u i ) i∈O and the blockchain swap protocol with sequential phases (σ P , σ T ) and let F T : {1, . . . , t T } → O ∪ {τ } be a decision function. The decision function F T is effective on the blockchain swap protocol (σ P , σ T ), if and only if the strategy profile (ŝ 1 , . . . , ŝn ) that specifies action 1 (follow the protocol) at any node is the unique subgame perfect equilibrium (in dominant strategies).

Proof. We have to prove the "if " since the "only if " directly follows from Proposition 3. By contradiction, if F T is not effective we have the following cases: (i)

P (v) = F T (l(v)) = b σ T 0 (A l(v)
), the original owner of the assets decides whether or not to follow the protocol or, (ii)

P (v) = F T (l(v)) = O k = b σ T 0 (A l(v)
) such that k ∈ {1, . . . , t T } : k = l(v), the activators are any player in the game but the original asset owners and the asset receivers.

(i) Whenever the owners of the assets in the initial configuration have to decide between action 0 (not follow the protocol) and action 1 (follow the protocol), by the second claim of Proposition 1 they have no incentive to follow the protocol σ T . That is, not following the protocol the asset set obtained at time t T is greater then the one obtained by following the protocol. (ii) Whenever the activators of transfers

(A k , O k ) are neither O k nor b σ T 0 (A k
), the situation is more complex. Assuming that players activate only exchanges of other players, it always exist an activator o l (v) : l(v) ∈ {1, . . . , t} that has to decide whether to follow the protocol σ T or to deviate by originating the sequence σ K T (defined in Section II-C). However, the activator would be indifferent to the two strategies since

u o l(v) (b σ T t ) -1 (o l(v) ) = u o l(v) (b σ K T t ) -1 (o l(v)
) due to the fact that the cardinality of the asset set remains unchanged.

Therefore, the strategy profile specifying action 1 (follow the protocol) at each stage cannot be a perfect equilibrium in dominant strategies if F T is not effective. This result seems to rule out any swap protocol with a non-effective decision function F T : {1, . . . , t T } → O. However, we can imagine protocols with transaction triggering mechanism not involving receiving agents (see Section III-B).

It is possible to have a subgame perfect equilibrium by adopting a penalty mechanism for protocol deviations. Protocols allowing the owner of the asset in the initial configuration to trigger the exchange have to contemplate a high penalty overcoming the value of the extra-asset gained in case of deviation. On the other hand, protocols with players activating exchanges not involving them may work with small penalties assigned in case of deviation. More precisely, by adopting a proper penalty mechanism, "following the protocol" still represents a subgame perfect equilibrium.

Corollary 1. If the decision function F : {1, . . . , t T } → O is not effective on a blockhain swap protocol (σ P , σ T ), then the strategy profile (ŝ 1 , . . . , ŝn ) that specifies action 1 at any node is the unique subgame perfect equilibrium (in dominant strategies) only when combined with a penalty function p :

A h → R + such that: • p(0; k) = ∈ R + for k = b σ 0 (A l(v) ) and, • p 0; b σ T 0 (A l(v) ) = δ + u O l(v) (b σ K T t T ) -1 (O l(v) ) -u O l(v) (b σ T t T ) -1 (O l(v)
) where δ ∈ R + . In the first case the activators are no more indifferent to the two strategies σ T and σ K T while in the second case the penalty matches, for a small δ, the advantage in deviating with respect to following the protocol σ T .

III. ATOMIC CROSS-CHAIN SWAP PROTOCOLS

This section is devoted to the analysis of existing crossblockchains swap protocols aiming to move forward the custodial trading performed by centralized exchanged services.

A. Sequential publishing and commitment

Here we present the swap solution proposed by Nolan [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] for permissionless UTXO-based blockchains. Nolan's protocol make use of contracts [START_REF] Wiki | Distributed contract[END_REF], hash-locks as commitment/locking scheme and time-locks to restore the initial situation consequently to failures in the publishing phase.

Given two asset owners (e.g., Alice and Bob) aiming at cross-swapping two crypto-assets (e.g., x Bitcoins and y Litecoins), the protocol (represented in Fig. 2) works as follows:

1) The agent Alice creates a secret s such that h = H(s), and publishes a contract transferring the ownership of her x Bitcoins to Bob on the Bitcoin blockchain. The contract is locked with the hashlock h and a timelock ∆ Bob ensuring that: "Bob can claim the asset property providing s before time ∆ Bob ". 2) When Bob confirms that Alice's contract has been correctly published on the Bitcoin blockchain, he publishes a contract on the Litecoin blockchain with the same hashlock h but with timelock ∆ Alice stipulating that: "before time ∆ Alice , Alice can claim the asset property with secret s". Note that ∆ Alice < ∆ Bob 3) When Alice confirms that Bob's contract has been correctly published on the Litecoin blockchain, she sends s to Bob's contract (before time ∆ Alice ), acquiring the y Litecoins and revealing s to Bob. 4) Bob then sends s (in the time interval [∆ Alice , ∆ Bob ]) to Alice's contract, acquiring the x Bitcoins and completing the swap. According to our formalization the two party cross-chain swap is represented as follows:

σ P = {(x, B), (y, A)}, F P (j) = {A, B}, j = {1, 2}; σ T = {(y, A), (x, B)} F T (k) = {A, B}, k = {1, 2}.
The protocol assumes that swap participants (i) actively monitor the involved blockchain in order to confirm contracts publication and, (ii) adopt a common hashing method. Herlihy [START_REF] Herlihy | Atomic cross-chain swaps[END_REF] extends the protocol for multi-assets and multi-agents swaps (with the secret creators forming a feedback vertex set L i.e., a subset of V whose deletion leaves D = (V, E) acyclic) and analyzes under which conditions it is 'atomic'. Atomicity, in this case, is defined in a game theoretical fashion: in [START_REF] Herlihy | Atomic cross-chain swaps[END_REF] a swap protocol is defined as atomic if:

-"following the protocol" is a nash equilibrium strategy and, -no conforming party is affected (in terms of payoff) by a protocol deviation. Moreover, multi-asset swaps that are atomic according to [START_REF] Herlihy | Atomic cross-chain swaps[END_REF] have a strongly connected corresponding digraph D (i.e., for every pair of distinct owners there is always a path from one to the other and viceversa).

1) The separating agent: In multi-players protocol implementations, the role of the secret creator or leader, denoted as l, becomes crucial. Let us consider only single leader swap protocols. The latter initiates the publishing and most importantly the commitment phase of the protocol by disseminating the secret s. A swap leader reveals the secret whenever all the contracts transferring her the ownership of the desired crypto-assets are correctly published. The sequentiality of the publishing together with the leader role ensures the required separation between the two phases of the blockchain swap. Let us formalize this concept in the following proposition: Proposition 5. A blockchain swap protocol verifying the atomicity definition proposed in [START_REF] Herlihy | Atomic cross-chain swaps[END_REF] and characterized by: (i) a leader participant initiating the publication phase F P (1) = {l} together with the commitment one F T (1) = {l} when all the contracts where she is directly involved are published, (ii) a sequential publishing phase where asset owners are called to publish as soon as all the contracts transferring them the desired assets' property are published, satisfies the commitment requirement of Definition 8.

Proof. We can state that the last contract(s) to be published are the ones involving the leader acting as a receiver; O t P = {l}. If a contract is not correctly published (∃ j ∈ {1, . . . , t P } : O j ∩ b 0 (A j ) = ∅), either the receiver is the leader, hence the commitment cannot start for (i), or the receiver is a different participant which for (ii) cannot publish her contracts. Indeed, since the graph is strongly connected, there is a sequence of participants from the receiver to the leader that will not activate their contracts.

2) Protocol strategic behaviour: The blockchain swap protocol presented above works with a sequential commitment phase where swap participants trigger contracts transferring them the ownership of the desired assets. Therefore, the decision function characterizing the protocol is effective. It should be noted that thanks to Proposition 4 we can derive a stronger result than the one proposed by the original paper concerning the players strategic behaviour.

Corollary 2. In the blockchain swap protocol (σ P , σ T ) presented in [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] (and generalised in [START_REF] Herlihy | Atomic cross-chain swaps[END_REF]), the strategy profile (ŝ 1 , . . . , ŝn ) specifying action 1 (follow the protocol) at any node is the unique subgame perfect equilibrium (in dominant strategies).

Let us provide (Fig. 3) the graphical representation of the extensive form game associated to the blockchain swap protocol (σ P , σ T ) presented in [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF]. We do consider two owners O = {A, B} swapping two assets A = {x, y} where t P = t T = 2. Alice, as leader of the swap protocol, is the first Subgame perfect equilibria are computed by applying the backward induction process. By reasoning from the end to the beginning of the game, at each decision step the strategy providing a better payoff (i.e., the one providing greater utility) is selected. Then, considering the game associated to the swap protocol presented in [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF], the sequence of optimal actions is the one specifying action 1 at each decision step (Fig. 3).

3) Atomicity violations: In [START_REF] Zakhary | Atomic commitment across blockchains[END_REF] the authors observe that the protocol presented by Nolan [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] is not immune to violation of the all-or-nothing atomicity as in Definition 7. More precisely, a time-lock expiration before commitment can lead an honest swap participant to deviate from the protocol. As in Fig. 3 if Bob does not commit the transfer at the last decision step (i.e., if the time-lock expires before) the outcome is b σ T 2 = (A, A) which differs from both b 0 and b * . Crash failures together with network delays are some of the possible causes of timelock expiration before commitment making the deviating party ending up with an asset loss. A snap commitment may be a solution to atomicity violations in asynchronous environments.

B. Concurrent publishing and snap commitment protocols

In [START_REF] Zakhary | Atomic commitment across blockchains[END_REF] authors propose an atomic swap protocol characterized by a concurrent publishing phase and a snap commitment. The protocol AC3TW works with a centralized trusted authority τ called Trent, acting as a separating agent that (i) verifies the correctness of the publishing phase and, (ii) witness the redemption contracts. Thanks to the trusted witness Trent the protocol benefits from all-or-nothing atomicity and faster publishing phase with respect to the sequential one (i.e., increasing overhead proportional to the number of contracts involved in the swap).

The protocol constructs for every possible swap configuration a directed graph D = (V, E) similar to the one of [START_REF] Herlihy | Atomic cross-chain swaps[END_REF] (see Section II-A for more details). D is multisigned by all swap participants in the set V generating a graph multisignature m s (D). The signatures order is irrelevant, the multisignature represents the participants' agreement on D.

Given two asset owners (e.g., Alice and Bob) aiming at swapping x Bitcoins for y Ethereum here below the steps characterizing the two party AC3TW protocol (represented in Fig. 4

σ P = {({x, y}, {A, B})}, F P (j) = {A, B}, j = {1}; σ T = {({x, y}, {A, B}, {X 1 (x) = B, X 1 (y) = A})} F T (k) = τ, k = {1}.
All-or-nothing atomicity is achieved by the fact that the redemption T (m s (D); RD) and the refund T (m s (D); RF ) events are mutually exclusive. The protocol meets the commitment requirement due to Trent's witnessing activity. The latter reduces as well the interactivity (i.e., the active participation of the swap participants) of the swap protocol [START_REF] Zamyatin | Xclaim: Trustless, interoperable, cryptocurrency-backed assets[END_REF] with respect to Nolan's swap implementation where asset owners have to be constantly on-line monitoring the involved blockchains. The two contracts of the AC3TW protocol are respectively a Bitcoin contract C 1 and an Ethereum smart contract C 2 (see [41; 42; 43] for more details) therefore, the protocol works in both UTXO-based and account-based blockchains [START_REF] Belotti | A vademecum on blockchain technologies: When, which and how[END_REF]. Moreover, concerning the protocol strategic behaviour we have that "following the protocol" is a Nash equilibrium. Proposition 6. Let N, (S i ) i∈N , (u i ) i∈N be the strategic form game associated with the swap problem A, O, b 0 , b * , (u i ) i∈O and the blockchain swap protocol (σ P , σ T ) characterized by a concurrent publishing and a snap commitment where the decision function F T is such that F T (k) = τ ∀k ∈ {1, . . . , t T }. Then, the strategy profile (ŝ 1 , . . . , ŝn ) that specifies action 1 (follow the protocol) for every player i is a Nash equilibrium. Proof. Given a strategy profile s = (s i ) i∈N ∈ j∈N S i whenever ∃i ∈ N such that s i is action 0 then, due to Trent witnessing, the protocol ends up in the original configuration b 0 . The outcome corresponding to the desired configuration b * is reached whenever action 1 is chosen by all the players. Hence, since u i (b -1 0 (i)) < u i (b -1 * (i)) ∀i ∈ N , action 1 is the dominant best response strategy to all s -i ∀i ∈ N . Fig. 5. Strategic form game associated to the blockchains swap protocol, presented in [START_REF] Zakhary | Atomic commitment across blockchains[END_REF]. Alice and Bob decide whether to publish or not the contract on the corresponding blockchain. Outcomes represent the payoffs of Alice and Bob respectively. By eliminating dominated strategies the emphasized outcomes are the equilibria of the game. The dominant strategy, the one providing strictly greater payoffs, is a Nash equilibrium.

Let us analyze the strategic form game associated to the blockchain swap protocol presented in [START_REF] Zakhary | Atomic commitment across blockchains[END_REF] in the case of two owners O = {A, B} swapping two assets A = {x, y} where t P = t T = 2. Strategic form games represent situations where players make decision simultaneously. In this cases, a matrix representation (Fig. 5 In order to identify the game's equilibria, dominated strategies (i.e., strategies providing a lower utility than others) have to be eliminated. Two different equilibria are computed: "following the protocol" is a dominant strategy always providing a greater utility for all the other players strategies, "do nothing" (i.e., choosing action 0) is a weakly dominant strategy that provides the same payoffs for all the other players strategies. Since dominant strategies are always Nash equilibria [START_REF] Peters | Game theory: A Multi-leveled approach[END_REF], the startegy profile specifying action 1 for every player of the game is a Nash equilibrium.

IV. CONCLUSIONS

This paper formalizes the distributed cross-chain swap problem in the blockchain context where parties exchange assets across multiple blockchains. To the best of our knowledge this work is the first to propose a complete framework allowing to analyze existing cross-chain swap protocols as strategic games. We prove that (i) following a swap protocol characterized by an effective decision function (e.g. the protocol proposed in [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] and generalised in [START_REF] Herlihy | Atomic cross-chain swaps[END_REF]) is a subgame perfect equilibrium in dominant strategies while (ii) following a swap protocol characterized by concurrent publishing and snap commitment (e.g. the protocol proposed in [START_REF] Zakhary | Atomic commitment across blockchains[END_REF]) is a Nash equilibrium. The presented framework can be further improved by considering a formal adversary model and concurrent commitment schemes (e.g., AC3WN [START_REF] Zakhary | Atomic commitment across blockchains[END_REF]). Our work opens several research directions. We are currently considering cooperative scenario where agents may collaborate to form coalitions. Analyzing the resilience of the cross-chain swap protocols to deviating coalitions is a challenging research direction. Another interesting open question is generalizing our framework in order to characterize protocols such as Lightning Networks.

, b σ k and b σ k- 1

 1 differ only for the ownership of assets A k belonging to b σ k-1 (A k ) at step k -1 and to O k at step k. Example 1. Let us consider the swap problem A, O, b 0 , b * , (u i ) i∈O such that the set of assets is A = {a, b, c, d, e}, the set of owners is O = {1, 2, 3}, the original ownership map is b 0 = (1, 1, 2, 3, 3) (meaning that the asset ownership is: 1 for assets a and b, 2 for c and 3 for d and e) and the desired ownership map is b * = (2, 3, 1, 2, 1).

  [29]. More precisely, vertexes are asset owners V = O and edges E e = (e i , e o ) : (e i , e o ) ∈ V 2 ∧ e i = e o can be derived by the original ownership map b 0 (providing e i ) and the desired one b * (providing e o ). Fig. 1 presents the digraph of Example 1.

Fig. 1 .

 1 Fig. 1. Digraph swap protocol representation of the sequence σ = ({a, c}, {1, 2}, {X 1 (a) = 2, X 1 (c) = 1}), ({b, e}, {1, 3}, {X 2 (b) = 3, X 2 (e) = 1}), ({d}, {2}, {X 3 (d) = 2}).

Fig. 2 .

 2 Fig. 2. Two party 'atomic' cross-chain swap protocol proposed in [16; 29] characterized by sequential publishing and commitment: σ P = {(x, B), (y, A)} and σ T = {(y, A), (x, B)}.

Fig. 3 .

 3 Fig. 3. Extensive form game associated to the blockchains swap protocol, presented in [16]. Publishing and commitment phases are sequentially executed one after the other. Outcomes represent the payoffs (assets) owned by Alice and Bob respectively. The figure represents the backward induction process for each decision step. player to decide between action 0 and action 1. Whenever one of the two parties opts for action 0 during the publishing phase, due to the commitment requirement (i.e., Definition 8), the games terminates with the original ownership configuration as the outcome: b σ P 1,2 = b 0 = (A, B) for A = {x, y}. If

Fig. 4 .

 4 Fig. 4. Two party AC3TW cross-chain swap protocol proposed in [26] characterized by concurrent publishing and snap commitment: σ P = {({x, y}, {A, B})} and σ T = {({x, y}, {A, B}, {X 1 (x) = B, X 1 (y) = A})}.

1 =

 1 ) allows to quickly analyze each possible outcome. In this case, Alice and Bob have to decide between action 0 and action 1 during the publishing phase. Once the transfers are correctly published (i.e., action 1 is chosen by both players) Trent commit the swap, b σ T b * = (B, A) for A = {x, y}. On the other hand, if one of the two parties chooses action 0 the outcome is b σ P 1 = b 0 = (A, B).

  ): (1) Alice and Bob create the digraph D and multisign it generating m s (D). (2) The multisignature is registered and stored by the centralized trusted authority, Trent, only if not registered before and it is set to a null value . (3) Alice publishes the contract C 1 on the Bitcoin blockchain stating that: -if Bob provides Trent's signature to the redemption instance, i.e., if he provides T (m s (D); RD) then, x Bitcoins' ownership is transferred from Alice to Bob. -if Alice provides T (m s (D); RF ) then, the x Bitcoins are transferred back to Alice. (4) Concurrently, Bob publishes a contract C 2 on the Ethereum network stating the following: -if Alice provides T (m s (D); RD) then, y Ethereums' ownership is transferred from Bob to Alice. -if Bob provides T (m s (D); RF ) then, the y Ethereums are transferred back to Bob. (5) After the publication of C i : i = 1, 2 either Alice or Bob requests Trent to trigger a redemption commitment scheme to redeem the assets. Trent issues T (m s (D); RD) only if both C 1 and C 2 are correctly published in their corresponding blockchains and the value of m s (D) stored by Trent is . (6) Whenever a contract is not correctly published any participant can request Trent to trigger a refund commitment scheme. T (m s (D); RF ) is issued only if m s (D) has value . (7) Depending on the case, Trent sets the value of m s (D) to T (m s (D); RD) or T (m s (D); RF ) accordingly. According to our formalization the two party cross-chain AC3TW protocol is represented as follows:


Claudia S Bianchini

Fabrizio Borgia
email: fabrizio.borgia@uniroma1.itchiadu14@gmail.comdalle@irit.frdemarsico@di.uniroma1.it

Patrice Dalle

Maria De Marsico

) Émergence

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION SCREENSHOTS SIGNWRITING (SW) SELECTED REFERENCES CORPUS MANAGEMENT TESTING

Fig. 1: The LIS sign for "various" written using SW.

SignWriting (SW) is a graphical framework which uses visual symbols (called glyphs) to represent both manual (configurations, movements, etc.) and non-manual (facial expressions, gaze, etc.) components of signs.

Researches of SLDS team in Rome have shown that SW can be used to properly represent sign languages. SW can be used either to write or to transcribe signed speech. SW alphabet currently lists about 37,000 glyphs.

SWIFT FEATURES

SWift allows digital representation of sign languages using SignWriting framework. It has been designed and tested for over one year in close contact with the SLDS team, which includes deaf researchers, therefore a true sample of the main target users. SWift's features are:

Web-based application, developed using XHTML, Javascript, PHP, ActionScript.

User-friendly interface, featuring:

• Minimization of the information provided by each screen

• Intuitive and familiar icons (triggering mouseover-animated clips for further explanation).

• Minimization of text labels

The user does not need to "learn" how to use the interface, he should rather "understand" it.

Glyph search engine, featuring an anatomical interface (Fig. 3) allowing the user to reach any glyph in few seconds, simply by choosing its traits (Fig. 4).

Glyph hint system, suggesting the user a set of compatible glyphs he might want to insert next, to save search time. Sign languages, similarly to many other world languages, have not developed their own writing system yet.

Most digital resources (such as the WWW) are only available in verbal language, which is difficult to acquire for the deaf.

A deaf person who wants to venture into the digital world must overcome several linguistic barriers, just like those he faces in everyday life! Any sign composed by the user can be saved on SWift's database, to aid in linguistic research. The sign is associated with its component glyphs and with some user-provided data (sign title, etc.). The frequency of occurrence of each involved glyph is incremented in the process. SWift's capabilities and usability were tested adapting the "Think-Aloud Protocol" in order to "fit" the needs of deaf users. The method has been named "Sign-Aloud Protocol".

CONCLUSIONS

An accurate design process, and a thorough test phase have granted SWift all the necessary features to become a widely-used SW editor. The application will help to weather the barriers keeping the deaf community out of the digital world.

Two development lines have been identified for the future of SWift:

• replacement of the current glyph search engine with an OCR-like recognition engine allowing the user to "free-hand" draw a glyph, which will be analyzed and recognized by the system • upgrade SWift with a semi-automatic image processing engine allowing the users to produce signs and to interact by gesture in front of a webcam

The participant is required to perform a list of tasks using SWift. Meanwhile, he is asked to sign anything that comes to his/her mind.

% users successfully decoded the interface 75 % interacted correctly with the glyph search engine

Fig. 2 :Fig. 4 :

 24 Fig. 2: Home screen of SWift, split in four areas of interest (highlighted by different colors).

Fig. 6 :

 6 Fig. 6: Spatial setting devised for SWift's evaluation.

Fig. 3 :

 3 Fig. 3: Anatomical interface used as starting point for the glyph search engine. The user can pick one anatomical area and search any glyph related to it using the interface shown in Fig. 4.

Fig. 5 :

 5 Fig. 5: Sketch of SWift's remote saving process.


Philippe Grandcolas

Ten false ideas about New Caledonia biogeography

The biogeographical paradigm of New Caledonia has recently changed. Although this island is now considered by many asoceanic, its study is still often impeded by some old misconceptions concerning either regional geology or phylogenetic analysisof evolution and biogeography. I discuss ten points that I feel are especially detrimental, to help focus on the real debate andthe real questions: (1) its geological history cannot be understood from the basement only; (2) the island submergence was notdue simply to sea-level variation; (3) Zealandia/Tasmantis is not a lost continent; (4) short-distance dispersal is not equivalent topermanence on land; (5) longdistance dispersal is not the sole event opposing vicariance, but short-distance dispersal as well; (6) the occurrence of relicts does not prove biota permanence; (7) a major fault system was not observed in New Caledonia; (8) terranes are not rafts; (9) forest climatic refuges do not necessarily equate to centres of endemism or centres of diversity; and (10)New Caledonia is not only a sink but also a source. Study of New Caledonia will need to focus on old and non-relict clades and there is a need to improve the local fossil record.

Islands have always fascinated biogeographers andtheir study has seen the emergence of several paradigms [START_REF] Darwin | On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life[END_REF][START_REF] Wallace | Island Life[END_REF][START_REF] Darlington | Zoogeography: the Geographical Distributionof Animals[END_REF][START_REF] Macarthur | The Theory of IslandBiogeography[END_REF]Nelson and Platnick,1981). These authors traditionally distinguished oceanic islands that were never connected to continents and emerged in the middle of oceans, as opposed to continental islands. The biotas of oceanic islands werenecessarily formed entirely by dispersal. In this context, opinions regarding New Caledonia have varied considerably. In early studies, New Caledonia was considered as an island intermediate between continental and oceanic and its biota was assumed to haveresulted from both vicariance and dispersal (Darwin,1859;[START_REF] Wallace | Island Life[END_REF][START_REF] Jeannel | La genèse des faunes terrestres: éléments de biogéographie[END_REF][START_REF] Faivre | Geographie de laNouvelle-Caledonie[END_REF][START_REF] Darlington | Zoogeography: the Geographical Distributionof Animals[END_REF][START_REF] Wilson | The nature of the taxon cycle in the Melanesianant fauna[END_REF]. In the second half of the twentieth century, some scientists came back to the idea of a continental island by presenting New Caledonia as a stunning Gondwanan refuge, based on its continental geological basement and the presence oflocal relict taxa [START_REF] Raven | Plate tectonics and Australasian paleobiogeography[END_REF]Holloway,1979;[START_REF] Stevens | Southwest Pacific faunal palaeobiogeography inMesozoic and Cenozoic times: a review[END_REF][START_REF] Lowry | Diversity, endemism, and extinction in the floraof New Caledonia: a review[END_REF]. This view prevailed for several decades until recent studies showed both its discordance with geology and the pitfalls of using relicts in biogeography [START_REF] Murienne | Evolution on ashaky piece of Gondwana: is local endemism recent in New Caledonia ?[END_REF][START_REF] Grandcolas | New Caledonia: a very old Darwinian island?[END_REF][START_REF] Grandcolas | Relict species: arelict concept?[END_REF][START_REF] Murienne | Testing biodiversity hypotheses in NewCaledonia using phylogenetics[END_REF]Swensonet al., 2014). In this way, several reviews showed thatmolecular phylogenies dated New Caledonian clades [START_REF] Grandcolas | New Caledonia: a very old Darwinian island?[END_REF][START_REF] Cruaud | Testing the emergence of New Caledonia: figwasp mutualism as a case study and a review of evidence[END_REF]Pillon,2012) to after the reemergence of the island, i.e. prob-ably 373 Ma according to geology [START_REF] Paris | Géologie de la Nouvelle-Calédonie. Un essai de synthèse (Mémoire pour servir de notice explicative à la carte géologique de la Nouvelle-Calédonie à l'échelle du 1/200000)[END_REF][START_REF] Aitchison | Eocene arc-continent collision in New Caledonia and implications for regional Southwest Pacific tectonic evolution[END_REF][START_REF] Cluzel | Tectonic accretion andunderplating of mafic terranes in the Late Eocene intraoceanicfore-arc of New Caledonia (Southwest Pacific): geodynamicimplications[END_REF]Cluzel et al.2012;[START_REF] Crawford | Chapter 25-120to 0 Ma tectonic evolution of the southwest Pacific andanalogous geological evolution of the 600 to 220 Ma TasmanFold Belt system[END_REF][START_REF] Pelletier | Geology of the New Caledonia region and itsimplications for the study of the New Caledonian biodiversity[END_REF]. Science is, however, not consensus and several papers have recently objected to the apparent congruence between systematics and geology by arguing for permanence of New Caledonian biota withad hoc scenarios of short-distance dispersal (SDD) among nearby islands [START_REF] Ladiges | New Caledonia-Australianconnections: biogeographic patterns and geology[END_REF]Heads, 2008a[START_REF] Heads | Biogeographical affinities of the New Caledonianbiota: a puzzle with 24 pieces[END_REF]. All these studies showed that New Caledonia is an interesting place for biogeography, and should allow us to disentangle the respective roles of vicariance anddispersal and open a very wide time window to test varied evolutionary scenarios. Some geological or biological inferences or theories already well established should be carefully specified for the debate could focus on the real controversial questions. Here, I review tenpoints that are frequently misunderstood and that I feel important to clarify for enhancing the quality offuture research on New Caledonian biogeography.

Its geological history cannot be understood from the basement only

The deepest geological basement of New Caledonia has a very old continental origin [START_REF] Paris | Géologie de la Nouvelle-Calédonie. Un essai de synthèse (Mémoire pour servir de notice explicative à la carte géologique de la Nouvelle-Calédonie à l'échelle du 1/200000)[END_REF]Pelletier,2007;[START_REF] Maurizot | New Caledonia geologicalmap[END_REF][START_REF] Cluzel | An outline ofthe Geology of New Caledonia; from Permian-MesozoicSoutheast Gondwanaland active margin to Cenozoic obductionand supergene evolution[END_REF]. However, many different layers and rock types have been overlaid on this basement in the context ofsubsequent and complex tectonic events. It is therefore confusing to describe New Caledonia simply as a piece of Gondwana [START_REF] Raven | Plate tectonics and Australasian paleobiogeography[END_REF]Holloway,1979;[START_REF] Stevens | Southwest Pacific faunal palaeobiogeography inMesozoic and Cenozoic times: a review[END_REF][START_REF] Lowry | Diversity, endemism, and extinction in the floraof New Caledonia: a review[END_REF] and not to take into account the consequences of more recent geological events. First, as emphasized by [START_REF] Murienne | Evolution on ashaky piece of Gondwana: is local endemism recent in New Caledonia ?[END_REF]and [START_REF] Grandcolas | New Caledonia: a very old Darwinian island?[END_REF], a recent and important submergence episode is documented by Palaeocene rock layers indicating deep marine sedimentation(thousands of metres) [START_REF] Paris | Géologie de la Nouvelle-Calédonie. Un essai de synthèse (Mémoire pour servir de notice explicative à la carte géologique de la Nouvelle-Calédonie à l'échelle du 1/200000)[END_REF]. This is the main evidence for submergence of the island: it does incorporate a few indirect assumptions, but merely strong stratigraphic and petrographic evidence interpreted inthe framework of the general regional tectonic model. Second, the ophiolitic nappe of Eocene lithospheric origin is also of particular significance. This geological structure has been known for a long time in relationto the discovery of nickel-rich soils and to the development of mining activities [START_REF] Paris | Géologie de la Nouvelle-Calédonie. Un essai de synthèse (Mémoire pour servir de notice explicative à la carte géologique de la Nouvelle-Calédonie à l'échelle du 1/200000)[END_REF]. Although it was considered as metamorphic or intrusive until the1970s [START_REF] Avias | Contribution à l'étude stratigraphique et paléontologique des formations anté crétacées de la Nouvelle-Calédonie centrale[END_REF][START_REF] Routhier | Etude géologique du versant occidental de la Nlle-Calédonie entre le col de Boghen et la Pointe d'Arama[END_REF], the presence ofthe ophiolitic nappe had already made it clear that important environmental changes had occurred in the past in New Caledonia. Before the development of plate tectonics theory, this nappe was supposed to have been overlaid gently on marine and then on terrestrial biota. It was then assumed that local groups adapted progressively to ophiolitic rocks and to the subsequent nickel-rich and unbalanced soils [START_REF] Morat | Floristic relationships of New Caledonian rainforest phanerogams[END_REF]. Since the 1990s, tectonics has developed integrative views based on several sources of data, from traditional historical geology and analyses of metamorphism to the study of boreholes. It was thus inferred that the ophiolitic nappe was overlaid byobduction of marine lithosphere (sea bottom) onto the island basement during a strong tectonic event, along with deep submergence of New Caledonia [START_REF] Paris | Géologie de la Nouvelle-Calédonie. Un essai de synthèse (Mémoire pour servir de notice explicative à la carte géologique de la Nouvelle-Calédonie à l'échelle du 1/200000)[END_REF][START_REF] Aitchison | Eocene arc-continent collision in New Caledonia and implications for regional Southwest Pacific tectonic evolution[END_REF][START_REF] Cluzel | Tectonic accretion andunderplating of mafic terranes in the Late Eocene intraoceanicfore-arc of New Caledonia (Southwest Pacific): geodynamicimplications[END_REF]Cluzel et al.2012;[START_REF] Crawford | Chapter 25-120to 0 Ma tectonic evolution of the southwest Pacific andanalogous geological evolution of the 600 to 220 Ma TasmanFold Belt system[END_REF][START_REF] Pelletier | Geology of the New Caledonia region and itsimplications for the study of the New Caledonian biodiversity[END_REF]. Geologi-cally, New Caledonia is actually a piece of Gondwana, but it has existed as a terrestrial island only since 37 Ma and it does not support any strong biogeographical hypothesis of a Gondwanan refuge. Speculating on the intensity and on the extent of the effectof the Eocene obduction on the New Caledonian landscape [START_REF] Ladiges | New Caledonia-Australianconnections: biogeographic patterns and geology[END_REF], that is complex to understand in details [START_REF] Crawford | Chapter 25-120to 0 Ma tectonic evolution of the southwest Pacific andanalogous geological evolution of the 600 to 220 Ma TasmanFold Belt system[END_REF]Pelletier,2007;[START_REF] Cluzel | An outline ofthe Geology of New Caledonia; from Permian-MesozoicSoutheast Gondwanaland active margin to Cenozoic obductionand supergene evolution[END_REF] will not save the Gondwanan refuge hypothesis, given the previous submergence episode that occurred already long after the separation from the remainder of Gondwana.

Submergence was not like sea-level variation

The paradigm of island submergence is sometimesseen as a minor geographical event that can be con-ceived by the mind of man, just like strong sea-level variation. In that line, some have assumed that Eocene mountain tops or parts of the island could have escaped submergence [START_REF] Morat | FloristicRelationships of New Caledonian Rain Forest Phanerogams[END_REF], as noticed by [START_REF] Murienne | Testing biodiversity hypotheses in NewCaledonia using phylogenetics[END_REF]. Although the mountain chains ofNew Caledonia are quite ancient (in contrast to those of New Zealand), the magnitude of the long tectonic events during the Palaeocene and Eocene was inferred to be so important that a large part of the basement was deep beneath the sea surface and partly beneath bottom-sea lithosphere. New Caledonia is a long and narrow island whose longest dimension is parallel to the borders of the tectonic plates. It was then strongly submitted to tectonic events. The island is remarkably well studied from a geological point of view and it is clear that it was deeply submerged during major tec-tonic events, the intensity of which needs to be properly understood by biologists [START_REF] Paris | Géologie de la Nouvelle-Calédonie. Un essai de synthèse (Mémoire pour servir de notice explicative à la carte géologique de la Nouvelle-Calédonie à l'échelle du 1/200000)[END_REF]Aitchisonet al., 1995;[START_REF] Cluzel | Tectonic accretion andunderplating of mafic terranes in the Late Eocene intraoceanicfore-arc of New Caledonia (Southwest Pacific): geodynamicimplications[END_REF][START_REF] Cluzel | An outline ofthe Geology of New Caledonia; from Permian-MesozoicSoutheast Gondwanaland active margin to Cenozoic obductionand supergene evolution[END_REF][START_REF] Crawford | Chapter 25-120to 0 Ma tectonic evolution of the southwest Pacific andanalogous geological evolution of the 600 to 220 Ma TasmanFold Belt system[END_REF][START_REF] Pelletier | Geology of the New Caledonia region and itsimplications for the study of the New Caledonian biodiversity[END_REF].

Zealandia/Tasmantis: not a lost continent

The name 'Zealandia' is increasingly used in the literature to characterize a large area of diverse continen-tal rocks that notably includes the basement of New Zealand and New Caledonia [START_REF] Luyendyk | Hypothesis for Cretaceous rifting of EastGondwana caused by subducted slab capture[END_REF]Mor-timer, 2004;[START_REF] Trewick | Hello NewZealand[END_REF]. The continental nat-ure of these rocks was known since long ago (Paris,1981), and until recently the area was known underthe name 'Tasmantis' employed both as regards geology [START_REF] Avias | A propos des "terres disparues" du Pacifique austral; tasmantis et Tasmantia[END_REF][START_REF] Lillie | The geology of New Caledonia[END_REF] and biogeography [START_REF] Smith | Molecular phylogeny of the scincid lizards of NewCaledonia and adjacent areas: evidence for a single origin of theendemic skinks of Tasmantis[END_REF]. However, naming such a geological diverse entity may be confusing for some biogeographers and lead them to assume that it could have behaved as a lost continent (i.e. a large terrestrial land mass some parts of which were flooded while others remained emerged), determining to shared distributions between terrestrial groups (e.g. [START_REF] Wanntorp | Past diversity ofProteaceae on subantarctic Campbell Island, a remote outpost ofGondwana[END_REF]. The lost continent paradigm can betraced back to the biblical story of the Deluge, employed in the earliest evolutionary scenarios fromthe Middle Ages to explain the occurrence of fossils: a large area of terrestrial land is supposed to be almost entirely flooded, determining a strong pattern of rela-tionships between the remaining emerged parts. This scenario implies naive assumptions predating tectonics: the area is supposed to be affected as a whole by a process combining relative terrestrial land stability and sea-level variation, implying flooding after a rise in thesea level or the collapse of the continent. In contradic-tion, tectonics showed from the mid-twentieth century that the landscape can be shaped by very strong events in a much more complex manner, including not only sea-level variation but also land subduction, uplift, orogenesis and subsidence occurring very locally and possibly in close opposition. The lost continent story recalls the long-standing myth of 'Pacifica' (Gregory,1930), briefly resurrected by some biogeographers [START_REF] Melville | Continental drift, Mesozoic continents and themigrations of the angiosperms[END_REF][START_REF] Nur | Lost Pacifica continent[END_REF]Nelsonand Platnick, 1981). Pacifica has been discounted [START_REF] Nunn | Vanished Islands and Hidden Continents of the Pacific[END_REF] but unfortunately Zealandia/Tasmantis remains fascinating, because New Zealand and NewCaledonia are traditionally considered as continental by many. Zealandia or Tasmantis is not a name for a past terrestrial geographical unit but for a complex series of rock layers that implies diverse emergent andsubmergent lands at different periods and after different events of subduction, uplift or sea-level variation,and ending today with a large ocean surface where more or less ephemeral islands are scattered.

SDD is not equivalent to local permanence

In the same line of reasoning, assumptions of SDD are often used to rescue vicariance scenarios. Following the confrontation between modern regional studies ofbiogeography and geology [START_REF] Murienne | Evolution on ashaky piece of Gondwana: is local endemism recent in New Caledonia ?[END_REF][START_REF] Grandcolas | New Caledonia: a very old Darwinian island?[END_REF], some authors realized thatstrict in situ permanence of biota has become an untenable hypothesis for New Caledonia. They then argued that hopping among ephemeral islands on the Norfolk Ridge could explain the permanence of old groups in New Caledonia [START_REF] Ladiges | New Caledonia-Australianconnections: biogeographic patterns and geology[END_REF]Heads,2008a). This scenario was envisaged by many studies asone among many possible others [START_REF] Morat | Floristic relationships of New Caledonian rainforest phanerogams[END_REF][START_REF] Swenson | Nothofagus biogeography revisited with special emphasis on theenigmatic distribution of subgenus Brassospora in New Caledonia[END_REF][START_REF] Grandcolas | New Caledonia: a very old Darwinian island?[END_REF] and it has been made testable by investigating whether the oldest inclusive insular Pacific clade is older than the oldest island, i.e. New Caledonia [START_REF] Nattier | Older than New Caledonia emergence? Amolecular phylogenetic study of the eneopterine crickets(Orthoptera: Grylloidea)[END_REF][START_REF] Swenson | Sapotaceaebiogeography supports New Caledonia being an old Darwinianisland[END_REF]. The congruence of this scenario with geology remains to be studied; vanished islandscertainly existed but were not suspected to be at theright place or the right time by geologists (Murienneet al., 2005;[START_REF] Grandcolas | New Caledonia: a very old Darwinian island?[END_REF][START_REF] Murienne | Testing biodiversity hypotheses in NewCaledonia using phylogenetics[END_REF].Regardless, it should be made clear that SDD is not equivalent to local permanence and cannot be taken as an argument to preserve the assumption of an anterior vicariance event between New Caledonia and adjacentso-called Gondwanan territories. Series of combined SDD events are mostly ad hoc hypotheses, the accumulation of which constitutes 'a house of cards' (Crispet al., 2011). They typically represent a heritage of the literature of the early ages of biogeography (Wallace,1881;[START_REF] Zimmerman | Distribution and origin of some eastern oceanic insects[END_REF][START_REF] Gressitt | Distribution patterns of Pacific island faunae[END_REF]Carlquist,1965;[START_REF] Axelrod | Ocean-floor spreading in relation toecosystematic problems[END_REF], and now need to be placed in a testable framework.

Dispersal (as opposed to vicariance) is either long orshort distance

There are not two sorts of dispersal, the "bad" one (LDD), speculative and incompatible with biogeo-graphic hypotheses, and the "good" one (SDD), that can be assumed without too many risks. All assumptions of dispersal need to be treated as such and to besubmitted to attempts of refutation. In this respect, [START_REF] Ladiges | New Caledonia-Australianconnections: biogeographic patterns and geology[END_REF] or Heads (2008a) misrepresented the situation as if LDD was opposed to SDD and vicariance for explaining the origin of the New Caledonian biota. In fact, the antagonist assumptionsareany kind of dispersal (SDD or LDD) against vicariance and extinction. We have emphasized that assuming vicariance between New Caledonia and, say, Australia implied assuming a permanence of terrestrial biota in New Caledonia [START_REF] Grandcolas | New Caledonia: a very old Darwinian island?[END_REF]. To employ the specific terminology of [START_REF] Heads | Bayesian transmogrification of clade divergencedates: a critique[END_REF], SDD was transmogrified into biotic permanence. Ad hoc dispersal assumptions have been extirpated from historical biogeography [START_REF] Nelson | Systematics and Biogeography: Cladistics and Vicariance[END_REF] and they should not now be reintroduced. Instead, it is better to refer to sister-group relationships, rather than misleadingly to permanence, SDD and vicariance.

Relicts do not prove the permanence of biota

Relicts are often taken as a conundrum (how dothey still exist?) or even as a strong indication that biota remain permanent (e.g. [START_REF] Ladiges | New Caledonia-Australianconnections: biogeographic patterns and geology[END_REF]Heads, 2008a). They are also often mistaken for odd concepts such as a living ancestor, primitive taxon or missing link [START_REF] O'hara | Telling the tree: narrative representation and thestudy of evolutionary history[END_REF][START_REF] Crisp | Do early branching lineages signifyancestral traits?[END_REF][START_REF] Crisp | Hypothesis testing inbiogeography[END_REF]. The reasoning given at length inseveral recent reviews needs not be repeated here in detail: it clearly states that relicts remain from old groups depleted by extinctions and are not necessarily old species; they need to be studied in a documented phylogenetic framework with the possible help of the fossil record [START_REF] Grandcolas | Relict species: arelict concept?[END_REF][START_REF] Grandcolas | What is the meaning ofextreme phylogenetic diversity? The case of phylogenetic relictspecies[END_REF]. The presence of Amborella trichopoda in New Caledonia, a unique species considered by many as the sistergroup of all other flowering plants [START_REF] Qiu | The earliest angiosperms: evidence from mitochondrial, plastid and nuclear genomes[END_REF], is one of the most emblematical examples of New Caledonian relicts. Unfortunately, although very impressive, this case remains inconclusive, as discussed in detail by [START_REF] Grandcolas | Relict species: arelict concept?[END_REF]: there is no informative fossil record for the monotypic genus Amborella that could allow us to disentangle the history of the very old stem-group from the history of the genus. Amborella trichopoda cannot be precisely dated; it may be very young as suggested by its recent local genetic differentiation [START_REF] Poncet | Phylogeography and niche modelling of the relict plant Amborella trichopoda (Amborellaceae) reveal multiple Pleistocene refugia in New Caledonia[END_REF] or much older as allowed by its very deep stem-group. Recently, another case study of supposed relict, the group of Beauprea plants (Proteaceae), confused thedating of the crown-group and of the stem-group and misleadingly argued for permanence of the lineage in New Caledonia [START_REF] He | Pre-Gondwanan-breakup origin of Beauprea (Proteaceae) explains its historical presence inNew Caledonia and New Zealand[END_REF]. Actually, relicts do not allow us to draw conclusions regarding phenotypic or geographical permanence; rather, they allow detecting large extinction events. In addition, assumptions of relicts and refuges are often twisted in a circular reasoning of mutual justification that needs to be broken [START_REF] Waters | Goodbye Gondwana? New Zealandbiogeography, geology, and the problem of circularity[END_REF].

Who has seen the fault?

It is difficult to refute the occurrence of something that has not been clearly observed nor explicitly described in detail. The case of supposed New Caledonia major fault given by Heads (2008b) is strange. Heads (2008b) referred to geological studies that didnot explicitly mention the fault or did not figure it. Most of the studies cited by Heads (2008b) are outdated regional syntheses and mentioned the fault under different concepts or names (sillon, furrow, line of weakness, fracture zone, geosuture, etc.), and often just in passing. Heads (2008b) himself recognized that local and recent field studies did not recognize it as a major feature along the entirety of New Caledonia [START_REF] Cluzel | Tectonic accretion andunderplating of mafic terranes in the Late Eocene intraoceanicfore-arc of New Caledonia (Southwest Pacific): geodynamicimplications[END_REF][START_REF] Chardon | Morphotectonic evolution of theNew Caledonia ridge (Pacific Southwest) from post-obduction tectono-sedimentary record[END_REF].The geological maps of [START_REF] Paris | Géologie de la Nouvelle-Calédonie. Un essai de synthèse (Mémoire pour servir de notice explicative à la carte géologique de la Nouvelle-Calédonie à l'échelle du 1/200000)[END_REF], the recently published Atlas of New Caledonia [START_REF] Bonvallot | Atlas de la Nouvelle-Calédonie[END_REF] and the recent geological map of New Caledonia available online [START_REF] Maurizot | New Caledonia geologicalmap[END_REF] did not show any major fault. The assumed distributional disjunctions (Heads, 2008b) along the unobserved fault represent small-scale distributional patterns that are often poorly documented and that can be interpreted within the framework of phylogenetic studies alongwith recent environmental change [START_REF] Murienne | Testing biodiversity hypotheses in NewCaledonia using phylogenetics[END_REF][START_REF] Nattier | Secondary sympatry caused by range expansion informs on the dynamics ofmicroendemism in a biodiversity hotspot[END_REF][START_REF] Nattier | Climate and soil type together explain the distribution of microendemic species in a biodiversity hotspot[END_REF][START_REF] Anso | Old lineage on an old island: Pixibinthus, a new cricket genus endemic to New Caledonia shed light on gryllid diversification in a hotspot of biodiversity[END_REF].

Terranes are not rafts

Another popular word in some recent biogeographi-cal studies is "terrane". And as with "Zealandia" and"dispersal", it has gained particular significance, apparently much further than justified by the actual evidence. Undoubtedly, present-day New Caledonia is an accumulation of rocks of different origins onto a continental basement [START_REF] Paris | Géologie de la Nouvelle-Calédonie. Un essai de synthèse (Mémoire pour servir de notice explicative à la carte géologique de la Nouvelle-Calédonie à l'échelle du 1/200000)[END_REF][START_REF] Maurizot | New Caledonia geologicalmap[END_REF]. However, the reasoning that different lineages were brought by different terranes (Heads, 2008a[START_REF] Heads | Biogeographical affinities of the New Caledonianbiota: a puzzle with 24 pieces[END_REF]) can be easily refuted [START_REF] Murienne | Panbiogeography of New Caledonia: a responseto Heads[END_REF]. From a biological point of view, tree topology or dating did not fit the theory of diverse origins by means of different terranes: the vast majority of local clades are more recent than 37 Ma and the so-called terranes. From a geological point of view, the terranes were emplaced during tectonic events the reconstruction of which implies most often non-biotic terrestrial conditions (see for example the ophiolitic nappe discussed in section "Its geologicalhistory cannot be understood from the basement only"). How could living things travel with lithosphere obducted onto the basement in the Eocene sea bottom? This appears to be a revival of the old demons of bio-geography; after the emphasis placed on the role of SDD, now comes the role of rafts.

Forest climatic refuges = centres of endemism = centres of diversity?

Along with the idea of relicts and escape from submergence often comes the concept of climatic forest refuges [START_REF] Pintaud | Chorology of NewCaledonian palms and possible evidence of Pleistocene rain forest refugia[END_REF][START_REF] Poncet | Phylogeography and niche modelling of the relict plant Amborella trichopoda (Amborellaceae) reveal multiple Pleistocene refugia in New Caledonia[END_REF][START_REF] Pouteau | New Caledonia: a Pleistocene refugium for rain forest lineages of relict angiosperms[END_REF]. The core idea conflating relicts and refuges is that some areas may be especially stable and hospitable and then shelter the remaining old lineages, an idea that is quite widespread in the literature [START_REF] Parsons | Environments and evolution: interactionsbetween stress, resource inadequacy and energetic efficiency[END_REF][START_REF] Grandcolas | Relict species: arelict concept?[END_REF][START_REF] Jordan | Palaeoendemic plants provide evidencefor persistence of open, well-watered vegetation since theCretaceous[END_REF][START_REF] Proches | Global hotspots in the present-day distribution of ancient animaland plant lineages[END_REF]. In this view, New Caledonia would be a kind of multi-million-year arch based on the combination of assumptions of a Gondwanan refuge and climatic stability in an oceanic insular context (for a similar view on New Zealand, see [START_REF] Gibbs | Ghosts of Gondwana: The History of Life inNew Zealand[END_REF]. The first part of the reasoning is flawed as the island was submerged repeatedly and the second part lacks continuity with the first part.

Positing areas of suitable climatic during the changes of the latest Quaternary may be possible and could elegantly explain some present-day distributions [START_REF] Pouteau | New Caledonia: a Pleistocene refugium for rain forest lineages of relict angiosperms[END_REF]. It does not mean, however, that such areas remain at the same place or of the same nature over millions of years, given that NewCaledonia moved from subtropical to tropical latitudesand that the regional climate has changed dramatically since the Eocene [START_REF] Reusch | New Caledonian carbon sinks at the onset of Antarctic glaciation[END_REF]. For example, [START_REF] Nattier | Secondary sympatry caused by range expansion informs on the dynamics ofmicroendemism in a biodiversity hotspot[END_REF] showed that the distribution area of microendemic species increased considerably with time, placing the oldest endemics in secondary sympatry. In this context, some Quaternary forest climatic refuges could be centres of endemism for some groups at the least, but they do not necessarily equate to centres of diversity or even centres of origin where a significant amount of deep time evolution is conserved [START_REF] Pellens | Assessing hotspots of evolutionary history with data frommultiple phylogenies: an analysis of endemic clades from NewCaledonia[END_REF]. As an example, the most emblematical relict in New Caledonia, Amborella trichopoda, considered by many as the sister-group of flowering plants, has a quite wide distribution on the island and is not restricted to particular areas of long-term stability [START_REF] Poncet | Phylogeography and niche modelling of the relict plant Amborella trichopoda (Amborellaceae) reveal multiple Pleistocene refugia in New Caledonia[END_REF][START_REF] Pouteau | New Caledonia: a Pleistocene refugium for rain forest lineages of relict angiosperms[END_REF].

Islands: sinks or sources, or both? Oceanic islands were often considered as sinks wherea few continental lineages established by dispersal (e.g. [START_REF] Macarthur | The Theory of IslandBiogeography[END_REF]. Continental islands were seen as refuges where old lineages remained, as emphasized by [START_REF] Gillespie | Arthropods on islands:colonization, speciation, and conservation[END_REF]. Both paradigms do not fundamentally envisage that islands can be sources for continents or other islands, but more or less ancient sinks. New Caledonia has been considered according to both these classical paradigms and therefore, until recently, never as a source. However, this situation changed as dispersal from islands back to continents recently became an attractive idea [START_REF] Bellemain | Are islands the end of thecolonization road?[END_REF] and a recent and elegant analysis revealed that New Caledonia and New Zealand have been both sinks and sources in the case of coniferous plants [START_REF] Condamine | Ancient islandsacted as refugia and pumps for conifer diversity[END_REF]. The change of paradigm is not fundamentally innovative as the most basic difference between a source and a sink is a scale effect: being larger and older, an area is expected to shelter more lineages with lower rates of extinction, and whose significantly higher ratio will be able to colonize other areas and to act as a source. The model behind the sink and source paradigm could take into account many other parameters, such as geographical pathways (not just dispersal), climatic equitability/stability and empty niches, that can differ significantly for islands. For New Caledonia, such a paradigmatic change is especially important as it helps in considering the island as otherwise than a Gondwanan refuge/sink. It also echoes the previous consideration that New Caledonia is not a museum but rather a cradle of diversity given the local diversification rates in several groups [START_REF] Espeland | Diversity dynamics in NewCaledonia: towards the end of the museum model?[END_REF].

Conclusions: what has to be done for New Caledonia? Listing these ten misleading ideas allows us to draw some general lines of research for the future. First, old ad-hoc assumptions in biogeography such as SDD and rafts must be made carefully testable; such assumptions have appeared recurrently in the literature and some recent views have repeated with few changes what can be found in Wallace (1881) without giving the appropriate credit. Clearly, the problem is not that such assumptions are formulated (to the contrary) but rather that they have rarely been properly tested. Second, to perform decisive tests, we need to carefully consider a relevant and accurate taxonomic or geographical sample (e.g. [START_REF] Nattier | Older than New Caledonia emergence? Amolecular phylogenetic study of the eneopterine crickets(Orthoptera: Grylloidea)[END_REF][START_REF] Swenson | Sapotaceaebiogeography supports New Caledonia being an old Darwinianisland[END_REF][START_REF] Anso | Old lineage on an old island: Pixibinthus, a new cricket genus endemic to New Caledonia shed light on gryllid diversification in a hotspot of biodiversity[END_REF][START_REF] Skipwith | Old but not ancient: coalescent species tree of New Caledonian geckos reveals recent post-inundation diversification[END_REF]. Sampling accuracy often needs to be improved by long field and taxonomic studies [START_REF] Najt | Zoologia Neocaledonica 4[END_REF][START_REF] Najt | Zoologia Neocaledonica 5.Systématique et endémisme en Nouvelle-Calédonie[END_REF][START_REF] Grandcolas | Zoologia Neocaledonica 6. Biodiversitystudies in New Caledonia[END_REF][START_REF] Grandcolas | Zoologia Neocaledonica 7. Biodiversitystudies in New Caledonia[END_REF][START_REF] Guilbert | Zoologia Neocaledonica 8. Biodiversity studies in New Caledonia[END_REF]. Third, geological knowledge needs to be confronted with biological analyses and not directly integrated within them to avoid circular reasoning [START_REF] Waters | Goodbye Gondwana? New Zealandbiogeography, geology, and the problem of circularity[END_REF]. This geological knowledge must be evaluated carefully with respect to the magnitude or the environmental significance of past events. Ideally, future studies should explore the whole regional time frame of evolution from the breakup of Gond-wana (>80 Ma) to the emergence of New Caledonia (37 Ma). This means focusing on old clades, but not necessarily on relict ones that are so depleted that they do not provide a sample of correct taxonomic diversity in space and time, except in the rare case of an accurate fossil record [START_REF] Grandcolas | Relict species: arelict concept?[END_REF]. The fossil record itself should not be considered as a backbone that is seldom available from the literature but rather should be actively improved upon to get direct information on the early distribution and characteristics of taxa. A rich local Quaternary record has already been studied inNew Caledonia [START_REF] Balouet | Fossil birds from late Quaternary deposits in New Caledonia[END_REF]Andersonet al., 2010;[START_REF] Worthy | Osteology supports a stem-galliform affinity for the Giant extinct flightless bird Sylviornis neocaledoniae (Sylviornithidae, Galloanseres)[END_REF] and an older and probably even richer record is available from the Cretaceous-Miocene [START_REF] Genise | Asthenopodichniumin fossil wood: different trace makers as indicators of different terrestrial palaeoenvironments[END_REF]R. Garrouste,A. Nel, P. Grandcolas, unpubl. data). It is only under these conditions that New Caledonia will become a natural laboratory comparable to other island systems.