GUEST EDITORS Primavera De Filippi

TOWARD A PHILOSOPHY OF BLOCKCHAIN INTRODUCTION

Keywords: blockchain, cryptocurrency, smart asset, smart contract, smart networks, ontology, epistemology, axiology, economic theory, algorithmic trust.</>

This article introduces the symposium "Toward a Philosophy of Blockchain," which provides a philosophical contemplation of blockchain technology, the digital ledger software underlying cryptocurrencies such as bitcoin, for the secure transfer of money, assets, and information via the Internet without needing a third-party intermediary. The symposium offers philosophical scholarship on a new topic, blockchain technology, from a variety of perspectives.

The philosophical themes discussed include mathematical models of reality, signification, and the sociopolitical institutions that structure human life and interaction. The symposium also investigates the metaphilosophical theme of how to create a philosophy of anything, specifically a new topic such as blockchain technology. Repeated themes are identified, in all areas of philosophical inquiry (ontology, epistemology, and axiology), and conceptual resources are elaborated to contribute to a philosophical understanding of blockchain technology. Thus, philosophy as a metaphilosophical approach is shown to be able to provide an understanding of the conceptual, theoretical, and foundational dimensions of novelty and emergence in the world, with a particular focus on blockchain technology.

Introduction and Background

What Is a Blockchain? Blockchain technology (a.k.a. distributed ledger technology) enables the secure transfer of money, assets, and information via the Internet without the need for a third-party intermediary, such as banks or other financial institutions (Swan 2015, ix). Transactions are validated, executed, and recorded chronologically in an append-only and tamper-resistant database, where they remain always available on the Internet around the clock for on-demand lookup and verification. Blockchain technology is what underpins such applications as the bitcoin cryptocurrency. Bitcoin was the first and perhaps most obvious application of blockchain technology-a decentralized payment system allowing for the real-time transfer of digital currency, at any time and to anyone in the world. Just as SMTP (simple mail transfer protocol) constitutes the underlying software protocol by which Internet users can send an e-mail to each other in a seamless and interoperable way, regardless of their e-mail provider, so the bitcoin protocol allows people to seamlessly transfer money between one another, regardless of their bank. With bitcoin, money can be transferred from one continent to another, at very low cost and in a matter of seconds, instead of waiting days or weeks and paying high commissions, as is the case with current international money transfers and remittance solutions.

But money transfer is just one application enabled by blockchain technology. The same technology also provides the means to record and transmit digital goods over the Internet, while ensuring that these goods cannot be copied or multiplied (thereby addressing the doublespending problem that has been an issue with digital currencies). Indeed, once they have been digitized as "smart assets," the recording, search, purchase, sale, tracking, and logging of resources can be coordinated with a much higher degree of automation, speed, trackability, and assurance. A blockchain can be used, for instance, as a digital registry to record, transfer, and verify asset ownership (home, auto, stocks, bonds, mortgages, and insurance) as well as to preserve the integrity and authenticity of sensitive documents or records (such as passports, visas, driver's licenses, birth and death certificates, voter registration, contracts, wills, patents, and medical records).

Eventually, as governments, corporations, and startups work toward the implementation of real-time payments, and also digital registration systems for the transfer and verification of digital assets and legal documents, a variety of legal, financial, and governmental services could be reengineered and readjusted for the Internet era.

Why the Philosophy of Blockchain?

The Internet is an important and transformative element that has arisen in the contemporary era.

Already in 2014, in Philosophical Engineering: Toward a Philosophy of the Web, Halpin and Monnin invited the discussion of the philosophical aspects of an emerging technology, the Internet [START_REF] Halpin | Philosophical Engineering: Toward a Philosophy of the Web[END_REF]. This is what inspired the framing of the present symposium, which considers philosophical themes in regard to another emerging technology, the blockchain.

The advent of blockchain technology brings about a new perspective or era in network computing, as indicated in figure 1. What we have witnessed thus far might be the first phase of the Internet, characterized by the transfer of information via simple networks. Today, a second phase of network computing is emerging-one that enables the secure, end-to-end, and computationally validated transfer of value (whether it is represented by money, assets, or contractual arrangements) via smart networks. The idea behind smart networks is that value transfer is performed by the network itself.

Intelligence is built directly into the network's operations through a sophisticated protocol that automatically identifies, validates, confirms, and routes transactions within the network. In the case of most existing blockchain-based networks, what makes the peer-to-peer transfer of value possible is the proof-of-work protocol (sometimes referred to as mining): that is, the economic competition between agents who contribute their computing resources to securing and maintaining the network. As a result of this process, a new form of "algorithmic trust" is created, one that significantly distinguishes itself from the more traditional typology of trust that was initially only between human agents.

One implication of transferring value with blockchain-based smart networks instead of relying on human-based institutions is that the traditional intermediaries responsible for verifying and validating transactions may become obsolete. As a result, the institutional structure of society could shift to one that is computationally based and thus has a diminished need for human-operated brick-and-mortar institutions. Blockchain technology applies to advanced and emerging economies alike. In the context of advanced economies, institutions could be reengineered, leading to a future where a substantial portion of human interactions, in particular value transfer and contractual engagements, are automated for quicker, easier, cheaper, and lower-risk execution. In the context of emerging economies, blockchain technology offers new opportunities for the two billion "unbanked" people in the world (PricewaterhouseCoopers 2016), and could serve as an important leapfrog technology in both monetary applications and digital asset registries. Land titling and property transfer systems have been stressed as having a crucial role in economic development ( [START_REF] Soto | The Mystery of Capital: Why Capitalism Triumphs in the West and Fails Everywhere Else[END_REF]. If widely adopted, blockchain-based networks could become a tracking register for the world's activity, a kind of societal memory (a concept explored in Greg Bear's Eon [1985]). Given the potential impact of blockchain technology in restructuring the traditional operation of economic, financial, legal, and governance systems, we can see great benefit in articulating a philosophy of blockchain as a conceptual resource for understanding these progressions in our modern world.

What Would Constitute a Philosophy of Blockchain?

The Internet prompted us to rethink such topics as the self, the relationship between the physical world and the virtual world, the individual and society, and the concepts of materiality, embodiment, temporality, spatiality, and possibility. Blockchains too warrant this degree of philosophical inquiry. They invite a full range of consideration in the classical areas of philosophy: ontology, epistemology, and axiology.

What Is the purpose of a Philosophy of Blockchain?

The task of philosophy is to help in naming, articulating, and describing different concepts and conceptual levels in their generative progression and meaning. Hence, the purpose of a philosophy of blockchain is to articulate conceptual resources for understanding what blockchain is and what it could be, its potential impact, advantages, and drawbacks, and the new opportunities it provides, both individually and societally. "Philosophy" here is used to refer to the theoretical underpinnings, the foundational definitions, as well as the general abstractions of blockchain technology, basically everything that might constitute the conceptual grounding of the notion of blockchain.

Such conceptual resources are necessary because blockchain technology brings together and integrates many fields, such as mathematical cryptography, distributed network technologies and versioning (for example, Git, Tor), financial accounting (ledgers, account balance instantiation and transfer), identity specification, economics and governance concepts, and userbased information security. Another reason these conceptual resources are needed is that blockchain is a complicated idea that is difficult to grasp immediately. Conceptual metaphors ("this is like that") can help us approach and comprehend new ideas [START_REF] Lakoff | Metaphors We Live By[END_REF].

As people first heard of blockchain technology as the software that underpins cryptocurrencies like bitcoin, the first conceptual metaphor was that "blockchain is like PayPal," an Internet-based way of transferring money from one person to another. Because of Bitcoin, blockchain seems to fit particularly well within the conceptual category of digital money and online payment systems.

The next conceptual metaphor draws upon network computing with statements like "blockchain is like Napster for money": a decentralized peer-to-peer network for transferring money on the Internet in a decentralized manner. In fact, in the extent that the bitcoin blockchain operates without human intermediaries, it challenges the monolithic and traditional understanding of the fundamental concept of money. How can we have money without a bank? Who confirms these transactions? Who maintains the centralized ledger recording all these transactions? Thus, the conceptual metaphor that "blockchain is like Napster for money" helps us see the blockchain as a peer-to-peer network without a central intermediary. Blockchain is like PayPal, but it also incorporates a few features that make it significantly different from the traditional PayPal model, in that it is an inherently decentralized system, operating on top of a peer-to-peer network, and without any central party in charge of coordinating the network.

These initial conceptualizations are based on a concrete and widely deployed use case (for example, the cryptocurrency bitcoin), but there is much more to that. Another conceptual metaphor currently in operation is that blockchain is a financial technology (fintech) capable of reengineering the financial services sector and enabling better and faster reconciliation between the accounts of different banks, securities brokers, traders, and clearing firms. This helps to provide the foundation for more generalized conceptualizations of blockchain technology that connote the greater capacities afforded by the technology, even if those have not yet been fully

realized. Yet another conceptual metaphor is that, at its core, blockchain technology is a nextgeneration network protocol that allows for the recording, transfer, and verification of physical or digital assets, both in the immediate term (spot market) and in future timeframes (futures market).

More generally, everything discussed so far falls under the conceptual metaphor of blockchain technology as a "better horse" (a better version of something we already know), as opposed to a "new car" (a disruptive innovation that gives rise to something completely new).

The latter applies to a whole category of applications that are as yet unplumbed-such as million-person genome databanks, global energy delivery and storage markets, and real-time voting systems [START_REF] Swan | Blockchain: Blueprint for a New Economy[END_REF]. For example, regarding the concept of money, the better horse notion of a blockchain is "digital cash" (something we already know), whereas the new car notion of a blockchain is "programmable money"-something that provides the ability to configure all of the parameters of a currency: who can use it, how, where, and for what, and what will happen in future time frames, such as expiration, redistribution, price indexing, and demurrage (the carrying costs associated with holding money). As opposed to digital cash, which simply represents money in a digital format, the ability to program the design and operations of money is a new capability that was not previously available. The digital realm also allows us to specify and compose our world (present or future) in new and exciting ways. Temporality is a particularly interesting property that becomes more configurable in the context of blockchainbased systems. The "blocktime" (the time over which a certain number of blocks will have confirmed) is a notion of time that is specific to a blockchain. Accordingly, there could be arbitrage opportunities between human clocktime cycles (for example, monthly rent is due) and blocktime cycles (for example, earning money in bitcoin or having a peer-to-peer bitcoin-based loan). Understanding blocktime as an alternative time paradigm could allow the contingency risk of future events to be reduced, for example, setting the temporality regime as a standard feature in smart contracts via a drop-down menu option (Swan 2016).

Symposium Thesis

The present symposium explores the metaphilosophical issue of whether an open call to thinkers to consider the philosophical implications of a new occurrence in the world-blockchain technology, in our case-can lead to the emergence of a new field of philosophy for that particular thing. The selected papers included here have proven our initial hypothesis to be true.

The call has given rise to a wide range of approaches in the contemplation of blockchain technology, and to the generation of conceptual resources that contribute to a wider understanding of blockchain technology as a new element of phenomenological reality and philosophical concern.

Summary of Contributions

This section provides a summary and analysis of the papers making up the symposium and of the broader philosophical themes they raise.

Summaries

The following are detailed summaries of the argument, method, conclusion, and implications of the findings in the seven ensuing papers.

(1) In "On the Philosophy of Bitcoin/Blockchain Technology: Is It a Chaotic, Complex System?" Renato P. dos Santos maintains that, while the bitcoin blockchain's proof-of-work consensus method is complicated, it is not complex. Building upon Crutchfield's Statistical Complexity, 1 dos Santos points to two principal methods for measuring the complexity of a system: algorithmic complexity and statistical complexity. The first method analyses the complexity of a particular data set by the length of the program necessary to reproduce such data. This method, however, is not generalizable to other data sets produced by the system or to the system as a whole. Accordingly, algorithmic complexity lacks the ability to distinguish between signal and noise. The second method, statistical complexity, relies on statistics as a means for measuring noise, and constitutes therefore a more general method of measuring complexity (Badii and Politi 1999, 9). 2 And so statistical complexity can be defined as measuring the amount of information in a system that describes the system's dynamics and that can be used to predict the future states of that system (Shalizi 2006, 59). Statistical complexity is measured along the parameters of randomness (entropy) and order (complexity). Applying Crutchfield's Statistical Complexity to measure the predictive information in the bitcoin blockchain consensus system (the process of cryptographically winning the opportunity to validate a new block of transactions), dos Santos finds a low degree of predictive information in the system, and he concludes that the system has entropy (randomness) but does not qualify as being complex. While the bitcoin blockchain consensus method has high entropy (per the randomness of the successful nonce guessing by any particular mining machine), on this formal measure it is not complex. The complexity-related characteristics of the bitcoin blockchain consensus method are more in line with those of a fair coin flip (Crutchfield's Statistical Complexity = 0), which has high randomness but also high order (low complexity); one will win the coin toss, just as one mining machine will win the nonce guess. On the other hand, the bitcoin blockchain consensus method's complexity-related characteristics are not in line with those of a double pendulum-with Crutchfield's Statistical Complexity >> 0 (much greater than 0)-which has a simple construction and less randomness but high disorder (high complexity), as it is unknown where the two pendula will be in any moment. The potential implication of this analysis is that blockchain systems-as noncomplex systems-would be unlikely to enter chaotic regimes, such as market flash crash situations.

(2) In "Blockchain Identities: Notational Technologies for Control and Management of Abstracted Entities," Quinn DuPont supports an understanding of digital technologies-such as blockchains-as "notional technologies" in the model of Nelson Goodman's "notational system," citing the blockchain-based digital artwork (photos and digital designs) asset management platform Monegraph as an example. Notational technologies are technologies that name and operationalize identities within a context. For Goodman, many situations count as notational schema (for example, the alphabet and natural language, where terms are indistinct, replicative, and confusing) but not more precisely as notational systems (in which terms are unambiguous and semantically disjoint and differentiated). Goodman's fine-grained model is helpful for specifying the distinct identities that arise in the context of a blockchain-based system. The notations in a blockchain are digital symbols that represent specific identities (such as a token, hash, or address). Identity management is a crucial element of digital systems, and the benefit of considering blockchains as Goodman-type notational systems gives access to the kind of granularity that might be necessary in specifying the detailed layers of identities in blockchain systems.

(3) In "Blockchain Axiology," Kobina Hughes argues that blockchain developers are in a unique position to address human rights concerns and reduce the digital divide. He suggests that, in addition to "blockchain lite" applications that generally address and improve commerce and governance, developers may also elect to focus on "blockchain heavy" applications that overtly target human rights protection in such areas as corruption and electoral fraud. For example, blockchain-based economic aid tracking applications could help to staunch capital flight, a serious global problem that at minimum impairs the ability of governance bodies to comply with the core U.N. human rights doctrine of the "responsibility to protect." Likewise, blockchainbased birth and death registries could be an immediate means of improving the documentation and tracking of individuals, particularly as related to human trafficking, refugee situations, and electoral fraud. The example of space exploration as a supranational "benefit of all" technology is drawn upon to create an inspirational vision for blockchain's capacity to create public goods.

The implication is that blockchain technology provides a more deliberate lever for addressing human rights issues than may have been available previously.

(4) In "Blockchain Technology as an Institution of Property," Georgy Ishmaev underlines the novelty of blockchain technology in the social context by considering a blockchain as a selfsufficient institution of "property." He suggests that blockchains have the necessary and sufficient conditions to compose the institution of property without reliance on traditional legal means. Invoking Penner's essentialist theory of property (which emphasizes exclusion and separability) and Hegel's recognitive system of property rights (which highlights universal access), Ishmaev shows how blockchains both eliminate the need for a third-party authority to enforce exclusion rights (a criterion raised by Penner) and provide a transparent system that facilitates the identification of existing property rights and increases people's ability to know and understand how the system works (a criterion imposed by Hegel). For Hegel, the existence of a complete model of property signals a far greater ontological existence than the one provided by the current regulatory framework of property rights. Today, a society-wide system that recognizes property would have to exist in order for any particular object to be recognized as property. In the context of a blockchain-based system, a set of cryptographically secured tokens can be used to indicate that there is a global system of cryptographically controlled assets.

According to Hegel's model, the very existence of a cryptographically activated asset also suggests the existence of a system for its registration, ownership, and transfer. And, indeed, these criteria are satisfied by the specific blockchain's protocol that provides universally accepted mechanisms for the registration and transfer of these tokens within the blockchain property system. The implications of these findings are that traditional property relations in society (that is, the means by which public property is currently administered) could eventually be replaced or supplemented by blockchain models and perhaps even be implemented in new domains, such as the ownership of private data (in virtual contexts).

(5) In "On the Continuity and Origin of Identity in Distributed Ledgers: Learning from Russell's Paradox," José Parra-Moyano suggests that in some cases of the setup and administration of smart assets (physical world assets controlled by blockchains) might give rise to a logical dilemma such as a Russell's Barber's Paradox-illustrated by the query, "If the barber is the one who shaves all those, and only those, who do not shave themselves, then does the barber shave himself?" Parra-Moyano introduces the notion of a "validating instance" as an entity that is required to bind an on-chain entity (such as a digital token or address) to an offchain asset (such as a physical world asset). In some situations, he argues, the mechanism might stall, because a validating instance cannot inscribe itself on a blockchain without another offchain entity entering into play in order to vouch for it-thus replicating the structure of a Russell's Barber's Paradox. After second thoughts, however, the problem does not seem to be a logical inconsistency as much as a loosely coupled control mechanism. If a validating instance is required to create a bridge between a blockchain-based asset and its representation in the physical world, then the key aspect is not whether or not an entity can inscribe itself in a blockchain but rather how the control coupling works between the on-chain and off-chain entities. One implication is that tighter coupling may be required between on-chain and off-chain assets if the smart assets concept is to be widely adopted as a virtual marshaling and control mechanism for assets in the physical world. Another implication is that logical analysis methods, such as known paradoxes, might be helpful as a critical tool with which to evaluate the security and other features of blockchain operations. A third implication is that computational systems might serve as a venue for novel mathematical discovery.

(6) In "Can Cyber-Physical Systems Reliably Collaborate Within a Blockchain?" Ben van Lier argues that, in an increasingly networked world comprising both virtual and physical systems, in which humans collaborate with machines, trust-producing mechanisms like blockchains will become crucial. Indeed, for van Lier blockchains are exemplary of the current state of the world as one composed of emergent cyber-physical systems (systems interconnecting virtual and physical components), with newly arising elements and properties merging and agglomerating into the existing world system to generate new wholes. The relevant and distinguishing properties of blockchains are fault-tolerant communication, voting, consensus, distributed ledgers, and transaction execution. This supports the idea that blockchains are an example of a technological phenomenon that is made up of different interconnected and autonomous systems that, when combined, create a new whole or entity in the world, both conceptually and operationally. According to van Lier, the future could be one world composed of many different networked "systems of systems," with multiple interconnected systems (virtual and physical) undertaking autonomous activities and making independent decisions. In a world made up of increasingly complex cyber-physical systems, complexity science could be an appropriate method for understanding and analysis of this new world. The implication of this analysis is a greater understanding of the nuances of how humans and machines may coexist cooperatively in a trust-based manner in the future.

(7) In "Computing Ledgers: The Political Ontology of the Blockchain," Pablo R. Velasco considers the political structure of monetary systems in the traditional sense and in the blockchain model, and he finds that blockchains trigger a shift in authority from institutions to computational systems. With blockchains, authority is displaced from the institutional actors in the system to the instrumental control of trust by the software. In order to understand the difference between fiat banking-system politics and distributed ledger politics, Velasco engages in a comparative analysis of the political ontology of money as it is traditionally produced by central banks and of money as it is produced in the context of a blockchain-based system. Both the traditional monetary system and a blockchain-based system are inherently political, in the sense that such qualities as control, trust, and authority are intrinsic to these two systems. Yet, while in fiat monetary systems, control, trust, and authority are vested in the institutional actors involved in the system, in blockchain-based systems, these qualities are properties of the software protocol. With a blockchain, the state disappears from the monetary realm: governmental authority is displaced not only from the money transmission process but also from the production process itself. This has at least two implications. One is that qualitative elements become established as part of the production process. The other is that as reality becomes increasingly composed of computer-made elements (such as blockchains, video games, and virtual reality), the production of qualitative goods (intangible social goods such as authority and trust) traditionally generated by human-operated institutions is being transferred to computational systems.

Philosophical Themes of Contributions

This section highlights, groups, discusses, and thematizes the philosophical aspects of the contributed papers. In Figure 3, there is a list of the papers and their philosophical themes. The papers in this symposium consider themes within three main categories of philosophical concern: ontology, epistemology, and axiology. The specific problematics addressed can be summarized as fitting models to the description of reality (ontology, is reality [papers 1, 5]), signification (epistemology, what can we know [START_REF]Other proposed methods for measuring complexity have not been sufficiently demonstrated-for example, power 2 Other proposed methods for measuring complexity have not been sufficiently demonstrated-for example, power laws and thermodynamic depth[END_REF]6]), and sociopolitical institutions (axiology, ethics, and right conduct in the world (3,4,7]).

The first theme, ontology, is fitting quantitative models to reality. Modernity is characterized by the application of scientific and mathematical models in an attempt to understand the world and new occurrences in reality. Two papers appeal to complexity science [START_REF]Also known as Crutchfield-Young Statistical Complexity (from 1989) and Grassberger-Crutchfield-Young Statistical Complexity[END_REF]6), and one invokes Russell's Barber's Paradox (5) in an attempt to comprehend blockchain as a new occurrence in the world, and they characterize its parameters. While the consensus algorithm of a blockchain may be complicated but not statistically complex on a formal definition ( 1), understanding a blockchain as an example of a heightened networked web of complex interactions between humans and machines may be a valid heuristic (6). Mathematical and logical formulations such as conjectures and paradoxes may also be helpful in identifying the structure of a new computational technology such as a blockchain (5).

The second theme, epistemology, is fitting models of knowing to reality. One paper explores signification and investigates the signifier-signified problematic [START_REF]Other proposed methods for measuring complexity have not been sufficiently demonstrated-for example, power 2 Other proposed methods for measuring complexity have not been sufficiently demonstrated-for example, power laws and thermodynamic depth[END_REF]. Another explores what we can know of network societies and human-machine collaboration (6). Other papers examine the correspondence between the physical and the virtual world [START_REF]Other proposed methods for measuring complexity have not been sufficiently demonstrated-for example, power 2 Other proposed methods for measuring complexity have not been sufficiently demonstrated-for example, power laws and thermodynamic depth[END_REF]5,6,7). Some of the key elements in the process of knowing include naming, the performing of actions, and the ability to confirm and refer to what is known. Epistemological issues are heightened in the blockchain context due to the coupling between the physical and virtual domain, where the tightness of linkage in these relationships is unconfirmed, both when they are initially established and persistently over time. One challenge in establishing correspondence between these two domains is due to the different natures of these two worlds: the virtual world is quantitative (digital ones and zeros), and the physical world is qualitative (messy, variable, irrational).

Ultimately, we are still in the early days of the experimental process to identify the computational equivalents of human-based qualities, such as trust and truth.

In the context of signification, the idea of using Goodman's notational system as a finegrained method for specifying blockchain identities suggests a broader analysis of signification.

For example, Saussure's dyadic semiotic sign (the signifier and the signified) and Peirce's sign (with explicit extension to the physical world) might be helpful in further delineating the correspondence of blockchain entities as they move between the physical domain and the virtual.

Performance is likewise important, including what counts as the performance of an action in the blockchain context, and what might grant identity or over-personify it. For humans, performativity is connected to the expression of identity. While this is not directly applicable to blockchains, one might consider performativity metaphorically. Both John Austin's speech acts (mere utterance has performative action) and Judith Butler's notion of performance (identity is real only to the extent that it is performed) can be transposed easily into the blockchain context in that transactions do not exist in a fully realized way until they are broadcast (that is, performed)-and if transactions are not broadcast to the network, they are simply deemed not to exist. Nuances in performativity theories could be helpful in further specifying the range of existence of blockchain entities; in the case of transactions, from wallet submission to validation, confirmation, broadcast, and propagation. Indeed, while a peer node merely publishing transactions on the network might not necessarily have a higher claim to identity than any other database, a smart contract that can update its own code might have more ground to do so.

Finally, when it comes to messages being received and acknowledged, blockchain networks have a sense of peer-to-peer reciprocity that is articulated in the "hear say yes" affirmation relation Derrida sees in Joyce (Derrida 1991, 256). Blockchain peer nodes engage in a "hear say yes" affirmation as the means by which state changes are confirmed and propagated through the system, since each party cosigns and assents to the new state of the network.

Accordingly, decentralization is not merely architectural (in the setup of the peer-to-peer mesh network) but also operational, in the sense that each participating node is responsible for evaluating and confirming the new blocks that will become part of the new state of the world, rather than just accepting the update from a hierarchically superior main node.

The third theme, axiology, is fitting models to sociopolitical institutions. Axiologically, From a high-level abstraction, the framing of all these papers is epistemological, in the sense that the general problem is a grasp at understanding blockchain technology as a new occurrence in the world. Within the general epistemological problem of an understanding of blockchains, there is a host of other issues: what blockchains are ontologically, what we can know of ourselves and our world through blockchains, and how we can make the world a better place (or not) axiologically with blockchains. To answer these questions, we must explore a variety of known ontological, epistemological, and axiological approaches to bring us toward a more profound articulation and understanding of blockchains, as has been demonstrated here with mathematical and scientific models, signification models (naming, performativity, and confirmation), and world sociohistorical political and aesthetic models.

Results and Conclusion

Implications of a Philosophy of Blockchain

The papers, both individually and in thematic synthesis, support the thesis of this symposium that conceptual resources may be obtained through an open call for the philosophical investigation of a topic, in this case blockchains. Overall, from the papers it emerges that, on one hand, blockchains seem to have some elements of models, concepts, and objects that we already know but, on the other hand, also have other elements that are unlike the things we know. This establishes blockchains as having an ontological status of novelty or emergence in providing something new compared to what we have previously understood as reality. The argument in favor of blockchains perceived as a novel occurrence in the world is configured in two ways in the papers. First, there is the idea of novel emergence in a unitary sense: blockchain technology is itself a new kind of thing. Second is the point that blockchains are a novel emergence in a generative and systemic sense: blockchain technology is not a stand-alone and discrete kind of thing but one that is intertwined with many other aspects of what we are doing and thinking about ourselves and reality, in both physical and virtual domains. In addition, blockchains connote the possibility of "moreness" for our existence in the world and our ability to shape and create reality. This line of philosophical reasoning suggests that a blockchain is not just a new widget but rather a new kind of widget. Taken to the extreme, blockchain technology might constitute a new and foundational mode of configuring reality. At minimum, blockchains could lead to a new era of network computing, in which the secure value transfer of money, assets, and contractual arrangements can occur in an automated and trustworthy manner via computational systems. It is as difficult now to foresee the full impact of blockchain technology as it was (and maybe still is) to estimate the full effect of the Internet.

While conceptual resources for a better understanding of blockchain technology have been obtained in this analysis, particularly in the sense of higher-order generalizations that explain the phenomenon, limitations must also be acknowledged. It would be pyrrhic to have the expectation of a singular and fully formed philosophy of blockchain as a result of this exercise.

In terms of the development of the field of philosophical inquiry, the philosophy of blockchain is a topic that is more conceptual than, for example, the formal scientific progression of ideas through the phases of conjecture, theory, hypothesis, and law. Although perhaps more than conjecture, the philosophy of blockchain does not yet qualify as a theory. This is because typically a theory is theoretically well supported, explanatorily powerful, and phenomenologically robust, with consistency and testability. On one hand, future efforts could work toward establishing a "theory of blockchain" in the philosophical domain, building on the ontological and epistemological claims made here. On the other hand, blockchain philosophy may never be formalized as a scientific theory, and there could be arguments against evaluating it this way. The key point is that some novel conceptual resources have been obtained in this exercise.

In conclusion, the primary result of this symposium is that conceptual resources have been generated for the problem at hand of achieving a better understanding of blockchain technology from a philosophical perspective. This constitutes a successful start toward building a comprehensive philosophy of blockchain to the extent that repeated themes have been identified, in all areas of philosophical inquiry (ontology, epistemology, and axiology), and ideas have been elaborated to contribute to a philosophical understanding of blockchain technology. The secondary result of this special issue is metaphilosophical. This is the validation of philosophy as a means for understanding a new domain, and as an investigative tool for the analysis of an emerging technology. Philosophy as a metaphilosophical approach is demonstrated as one that can provide an understanding of the conceptual, theoretical, and foundational dimensions of novelty and emergence in the world, particularly in this case of the emergence of blockchain technology. The immediate benefit of substantiating philosophy as an approach for blockchain studies is that it helps to frame the field of blockchain philosophy. The positivist ontological and epistemological claims made in this analysis could lead to more specific inquiry in areas ranging from the foundational to the applied. New areas of investigation might be inaugurated-for example, critical blockchain studies, the economic theory implications of blockchains, and an examination of temporality, risk, and futurity in decentralized financial systems.

Melanie Swan

Primavera de Filippi

Figure 1 .

 1 Figure 1. Two fundamental eras of network computing (expanded from Sigal 2011)

  some of the main topics regarding what is valorized or ignored in the world are ethical and moral issues, and aesthetics. First looking at what we valorize as being moral or ethical, we are concerned with right conduct as we enact our lives individually and collectively. In this area, one paper looks at property as a political institution (4), another at the political structures of money production (7), and a third at human rights issues in the context of political corruption (3).Second, concerning what we valorize as aesthetic (what is elegant, efficient, beautiful, or sublime), one paper discusses the blockchain-based protection of digital artworks[START_REF]Other proposed methods for measuring complexity have not been sufficiently demonstrated-for example, power 2 Other proposed methods for measuring complexity have not been sufficiently demonstrated-for example, power laws and thermodynamic depth[END_REF], with the assumption that digital artworks are valuable in society. A broader consideration of aesthetics of blockchain might encompass on one hand an evaluation of the computational elegance or efficiency of the cryptographic equations or software, for example using techniques like Occam's razor (the simplest solution is the most efficient and elegant). On the other hand, the aesthetics of blockchain might range to the analysis of artworks made with blockchain code or the political messaging of blockchains, such as Cryptoart's fine art combined with bitcoin storage or cryptographic art, such as the ASCII Bernanke recorded in the bitcoin blockchain[START_REF] Pastebin | ASCII Bernanke[END_REF].

What behavioral norms are being established? Who is adopting blockchain technology and why? What constitutes an aesthetics of blockchain in the sense of what aspects are seen and valued as being beautiful, elegant, or aesthetically pleasing?

  Broadly speaking, ontology treats questions of existence. What is blockchain technology? How is it being characterized, created, and implemented? How does it operate in the world? What are its definitions, classifications,

	teleology, possibilities, constraints, and limitations? Accordingly, from a practical point of view,
	an ontological philosophy of blockchain would provide a concise definition of what the
	technology is, including its purpose, function, and dimensions. The second area, epistemology,
	deals with knowledge. We can ask what new kinds of things blockchain technology is helping us
	to know or understand. This can be with regard to both new knowledge and new ways of
	knowing. We can ask about the corresponding standards of proof, or truth, that supports this new
	knowledge. Is there new knowledge that is required to create and engage with blockchain
	technology? The third area, axiology, especially ethics and aesthetics, concerns how blockchain
	technology is valorized, taken up, and regarded by individuals and society. What aspects are
	being valued, overvalued, undervalued, and overlooked, and by whom? What is being omitted
	and why?

Table 1 .

 1 List of papers and philosophical themes

	Paper title

Acknowledgments

A special thank you to Renato P. dos Santos, Florian Glaser, and Craig Hayenga for reviewing certain sections.


Pierre Lescanne
email: pierre.lescanne@ens-lyon.fr

RR2006-51 (version 3) CP games: a tutorial

Introduction

This paper aims to be a didactic introduction to concepts developed in [START_REF] Le Roux | A discrete Nash theorem with quadratic complexity and dynamic equilibria[END_REF] and used in [START_REF] Chettaoui | Rewriting game theory as a foundation for state-based models of gene regulation[END_REF] as a formalization of gene regulation networks. CP games rely on players and synopses that are also game called situations. A player may change a synopsis to another provided the synopsis can be converted to the other, moreover a synopsis can be preferred to another. Conversion and preference are two basic concepts of CP games from which they take their name (C and P). The following tutorial presents all those concepts through examples.

Basic concepts

To start with, let us give the two main notions of games. First, with no surprise, a game involves players. Second, a game is characterized by situations. In CP games these situations will be called synopses or sometimes game situations. A player can move from one synopsis to another, but she 1

1 See the preface of [START_REF] Osborne | An Introduction to Game Theory[END_REF] for the use of personal pronouns does that under some constraints as she has no total freedom to perform her moves, therefore a relation called conversion is defined for each player; it tells what moves a player is allowed to perform. Conversion of player Alice will be written Alice . As such, conversion tells basically the rules of the game. In chess it would say "a player can move her bishop along a diagonal", but it does not tell the game line of the player. In other words it does not tell why the player chooses to move or how to convert her synopsis. Another relation called preference compares synopses in order for a player to choose a better move or to perform a better conversion. Preference of player Beth will be written Beth and when we write s Beth s we mean that Beth prefers s to s or, rather than s, she chooses s . Preference (or choice) is somewhat disconnected from conversion, a player can clearly prefer a synopsis she cannot move to and vice versa she can move to a synopsis she does not prefer.

A key concept in games is this of equilibrium. As a player can convert a synopsis, she can convert it to a synopsis she likes better, in the sense that she prefers the new synopsis she converted to. A player is happy in a synopsis, if there is no synopsis she can convert to and she prefers. A synopsis is an equilibrium if each player is happy with this synopsis. We will see that this concept of equilibrium captures and generalizes the concepts known as Nash equilibrium in strategic games .

Some examples

Let us present the above concepts of conversion, preference and equilibrium through examples. We will introduce a new concept called change of mind.

A simple game on a square

As an introduction, we will look at variations of a simple game on a board.

A first version

Imagine a simple game where Alice and Beth play using tokens on a square. We number the four positions as 1, 2, 3 and 4.

Assume that player Alice has a red round token and that player Beth has a blue squared token. The two player places their tokens on one vertices and then they move along edges. They can also decide not to move. Assume that Alice and Beth never put their token on a vertex taken by the other player, more precisely a move towards an occupied vertex corresponds to a capture of the token at that vertex and to a win. Then the game ends. The game has 12 situations or synopses, which we write i|j for 1 ≤ i, j ≤ 4 and i = j. The above pictured situation corresponds to 1|2. The two conversions are described by Figure 1 In this game, both players share the same preference, namely the following: since a player does not want her token to be captured, she prefers a synopsis where her token is on the opposite corner of the other token to a synopsis where her token is next to the other token. This gives the preference given in Figure 1 right. The arrow from 1|2 to 1|3 means players prefer 1|3 to 1|2.

4|2 u j r v D R @ H n v 4|3 4|1 1|2 3|2 1|3 B P v D R k s ! k s s 3|1 l t u j r C Q w C Q Ù 1|4 3|4 2|3 2|1 2|4 l t B P F T t h p v 4|2 x x E E q q w w x x 4|3 4|1 1|2 3|4 1|3 k k s s q q o o m m o o 3|1 x x C C Q Q E E G G I I G G 1|4 3|2 2|3 2|1 2|4 I I m m
left. In this figure, C Q is Alice's conversion and C Q is Beth's conversion. 4|2 x x E E q q 4|3 4|1 1|2 3|2 1|3 n n p p 3|1 x x F F H H 1|4 3|4 2|3 2|1 2|4 I I m m 4|2 x x # | d 9 f E E d m | # ) q q f I 9 3 4|3 4|1 1|2 3|2 1|3 # | d 9 f n n x f S D 7 p p h p | Ù % 3|1 9 f x x # | d F F x f S D 7 H H h p | Ù % 1|4 3|4 2|3 2|1 2|4 9 f # | d I I f I 9 3 m m d m | # )
From the conversion and the preference we build a relation that we call change of mind. Alice can change her mind from a synopsis s to a new one s , if she can convert s to the new synopsis s and rather than s she chooses s . Changes of mind for Alice and Beth are given in Figure 2 on the left. In this figure, G G is Alice's change of mind and G G is Beth's conversion. The (general) change of mind is the union of the agent change of mind, it is given by Figure 2 on the right. The equilibria are the sinks (or "minimal point") for that relation, namely 1|3, 4|2, 3|1 and 2|4. This means that no change of mind arrows leave those nodes. In these situations players have their tokens on opposite corners and they do not move. An equilibrium like 1|3 which is a sink is called a Abstract Nash Equilibrium.

A second version

We propose a second version of the game, where moves of the token can only be made clockwise. Obviously the conversion changes, but also the preference, as a player does want not be threatened by another token placed before her clockwise and prefers a synopsis that places this token as far as possible. The conversions, the preferences and the changes of mind are given in Figure 8 (page 21). If one looks at the equilibrium, one sees that there is no fixed position where players are happy. To be happy the players have to move around for ever, one chasing the other. It is not really a cycle, but a perpetual move. We also call that an equilibrium. It is sometime called a dynamic equilibrium or a stationary state according to people you talk to.

A third version

The third version is meant to present an interesting feature of the change of mind. In this version, we use the same rules as the second one, except that we suppose that the game starts with both token on the board or it starts as follows. Alice has put her token on node 1 (this game positions is described as 1|ω). Then Beth chooses a position among 2, 3 or 4. The conversion is given in Figure 3 left. Beth may choose not to play, but in this case she looses, in other words, she prefers2 any position to 1|ω. The change of mind is given on Figure 3 right (page 7). There is again a dynamic equilibrium and one sees that this dynamic equilibrium is not the whole game, indeed one enters the perpetual move after at least one step in the game.

Strategic games

In this presentation of strategic games we do not use payoff functions, but directly a preference relation3 and we present several games.

The Prisoner's Dilemma

The problem is stated usually as follows Two suspects, A and B, are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal: if one acts as an informer against the other (finks) and the other remains quiet, the betrayer goes free and the quiet accomplice receives the full sentence. If both stay quiet, the police can sentence both prisoners to a reduced sentence in jail for a minor charge. If each finks, each will receive a similar intermediate sentence. Each prisoner must make the choice of whether to fink or to remain quiet. However, neither prisoner knows for sure what choice the other prisoner will make. So the question this dilemma poses is: What will happen? How will the prisoners act?

4|2 k s v m u 4|3 4|1 1|2 3|2 1|ω 4 B C Q R 1|3 C Q ! k s 3|1 k s t w C Q 1|4 3|4 2|3 2|1 2|4 C Q E S v 4|2 o o $ { 8 g { $ 0 q q g 8 2 4|3 4|1 1|2 3|2 1|ω 8 8 Ù j x G G S S Q ` d k 1|3 $ { G G 8 g g C 6 o o f { & 3|1 o o 8 g x x $ { g C 6 G G f { & 1|4 3|4 2|3 2|1 2|4 8 g G G $ { I I g
Each prisoner can be into two states, either fink (F ) or be quiet (Q). Each prisoner can go from Q to F and vice-versa, hence the following conversion, where C Q is prisoner A conversion and C Q is prisoner B conversion (Figure 4 left). Each prisoner prefers to go free to been sentenced and prefers a light sentence to a full sentence. Hence the preference are given in Figure 4 right,where G G is prisoner A preference and G G is prisoner B preference.

From this we get the change of mind of Figure 5. One sees clearly that the only equilibrium is F, F despite both prefer Q, Q as shown on Figure 4 right.

Such an equilibrium is called a Nash equilibrium in strategic game theory. The paradox comes from the fact that F, F is an equilibrium despite the fact one has:

Q, Q k s C Q Q, F u F, Q u F, F C Q k s Q, Q G G g g Q, F w w r r F, Q U U s s F, F G G l l
Q, Q G G Q, F F, Q F, F G G Q, Q G G Q, F 1 1 1 F, Q 1 1 1 F, F G G
Q, Q s s k k F, F in the preference.

Matching Pennies

This second example is also classic. This is a simple example of strategic game where there is no singleton equilibrium.

The game is played between two players, Player A and Player B. Each player has a penny and must secretly turn the penny to heads (H) or tails (T ). The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), Player A wins. If the pennies do not match (one heads and one tails), Player B wins.

The conversion is similar to this of the prisoner's dilemma (Figure 6 left) and the preference is given by who wins (Figure 6

center).

Change of mind for matching pennies is in Figure 6 right. One notices that there is cycle. This cycle is the equilibrium. No player has clear mind of what to play and changes her minds each time she looses.

Scissors, paper, stone

Here we present the famous game known as scissors, paper, stone. It involves two players, Alice and Beth who announce either scissors (C) or paper (P ) or stone (T ) with the rules that stone beats scissors, scissors beat paper, and paper beats stone. There are nine situations (see below), one sees that Alice may convert her situation P, C to P, P or P, T and the same for the other situations. The conversion is given below left. Since the rules, it seems clear that Alice prefers T, P to P, P and P, P to C, P , hence the preference given below right with G G is Alice's preference and G G is Beth's preference. To avoid a cumbersome diagram, in the preference we do not put the arrows deduced by transitivity.

H, H k s C Q H, T u T, H u T, T C Q k s H, H G G Ù Ù H, T r r T, H y y D D T, T o o s s H, H G G H, T 1 1 1 T, H y y 1 1 1 T, T o o
C, C p x 8 F g u ' C, P C Q k s Ó C, T Ó C Q k s P, C u p x 8 F P, P C Q k s u P, T C Q k s u T, C f n H V u T, P u C Q k s T, T C Q k s u C, C t t 5 5 q q 0 0 C, P G G l l C, T Ð Ð P, C P, P G G l l s s P, T G G r r T, C j j Y Y T, P s s T, T G G r r s s
From the above conversion and preference, one gets the following change of mind.

C, C

t t f q q " & 1 6 @ E C, P G G G G o o C, T E @ 6 1 & " y y 1 1 1 P, C 1 1 1 y y 1 1 1 P, P G G 1 1 1 P, T G G R R f T, C j j f o o T, P 1 1 1 × × " & 1 6 @ E T, T G G 1 1 1
One sees also perpetual moves as in the matching pennies of which it is a generalization.

Strategic games as CP games

A strategic game is a specific kind of CP games. To be a strategic game, a CP game has to fulfill the following conditions.

1. Each synopsis is a n-Cartesian product, where n is the number of players. The constituents of the Cartesian product are called strategies.

2. Conversion for player a, written a , is any change along the a-th dimension, i.e., (s 1 , ..., s a , ..., s n ) a (s 1 , ..., s a , ..., s n ) if and only if s a = s a . Hence in strategic games, conversion is

• symmetric, (s a s implies s a s),

• transitive, (s a s and s a s imply s a s ),

• and irreflexive (it is not true that s a s).

Blink or you loose

Blink and you loose is a game played on a simple graph with two undifferentiated tokens. There are three positions:

GFED @ABC •• GFED @ABC GFED @ABC • GFED @ABC • GFED @ABC GFED @ABC ••
There are two players, Left and Right. The leftmost position above is the winning position for Left and the rightmost position is the winning position for Right. In other words, the one who owns both token is the winner. Let us call the positions L, C and R respectively. One plays by taking a token on the opposite node.

A first tactic: Foresight

A player realizes that she can win by taking the opponent's token faster than the opponent can react, i.e., player Left can convert C to L by outpacing player Right. Player Right, in turn, can convert C to R. This version of the game has two singleton equilibria: L and R. This is described by the following conversion

L C k s C Q R where C Q is the conversion for Left and C Q is the conversion for Right. The preference is L u u C u u S S R S S
where G G is the preference for Left and G G is the preference for Right. The change of mind is then:

L C o o G G R
and one sees that there are two equilibria: namely L and R, which means that players have taken both token and keep them.

A second tactic: Hindsight

A player, say Left, analysis what would happen if she does not act. In case Right acts, the game would end up in R and Left looses. As we all know, people hate to loose so they have an aversion for a loosing position. Actually Left concludes that she could have prevented the R outcome by acting. In other words, it is within Left's power to convert R to C. Similarly for player

Right from L to C. L C Q C R k s
We call naturally aversion the relation that escapes from positions a player does not want to be, especially a loosing position. Aversion deserves its name as it works like conversion, but flies from bad position. We get the following change of mind:

L G G C R o o
with C the singleton equilibrium

A third tactic: Omnisight

The players have both hindsight and foresight, resulting in a C/P game

L I W C q y I W R q y
with one change-of-mind equilibrium covering all outcomes thus, no singleton equilibrium exists.

L A A i C i i i A A i R i i i

The λ phage as a CP game

The λ phage is a game inspired from biology [START_REF] Ptashne | Genetic Switch: Phage Lambda Revisited[END_REF][START_REF] Ptashne | Genes and Signals[END_REF]. The origin of the game will be given in Section 5, here we give just the rules of the game.

There are three players cI, cro and Env. The game can be seen as a game with two tokens moving on two graphs where each player may choose to move one of the two tokens 4 . Env hates to see both tokens on the bottom position and moves one of them. The conversion is therefore the same for the three players 5 and is given by the following rightmost diagram:

cI cro cI 2 , cro 0 k s C Q u cI 2 , cro 1 u cI 1 , cro 0 k s C Q u cI 1 , cro 1 u cI 0 , cro 0 k s C Q cI 0 , cro 1
The preference is difficult to describe as an actual game to be played, it comes from the genetics and is specific to each player. The philosophy is as follows: a gene prefers a position if it is "pushed forward" that position.

cI 2 , cro 0 cI 2 , cro 1 cI 1 , cro 0 cI 1 , cro 1 cI 0 , cro 0 cI 0 , cro 1 cI 2 , cro 0 cI 2 , cro 1 ⊃ ⊃ cI 1 , cro 0 cI 1 , cro 1 ⊃ cI 0 , cro 0 cI 0 , cro 1 cI 2 , cro 0 cI 2 , cro 1 cI 1 , cro 0 cI 1 , cro 1 cI 0 , cro 0 cI 0 , cro 1
From the conversion and the preferences one deduces three changes of mind.

cI 2 , cro cI 2 , cro 1 cI 1 , cro cI 1 , cro 1 cI 0 , cro cI 0 , cro 1 cI 2 , cro 0 cI 2 , cro 1 ⊃ cI 1 , cro 0 cI 1 , cro 1 ⊃ cI 0 , cro 0 cI 0 , cro 1 cI 2 , cro 0 cI 2 , cro 1 cI 1 , cro 0 cI 1 , cro 1 cI 0 , cro 0 cI 0 , cro 1
from which we deduce the (general) change of mind of the game:

cI 2 , cro 0 cI 2 , cro 1 o oo o cI 1 , cro 0 cI 1 , cro 1 o oo o cI 0 , cro 0 y y y y G G G G cI 0 , cro 1
One sees one singleton equilibrium namely cI 0 , cro 1 (called the lyse) and one dynamic equilibrium namely { cI 2 , cro 0 , cI 1 , cro 0 } (called the lysogen).

Presentation of C/P games

To define a C/P game we have to define four concepts:

• a set A of agents,

• a set S of synopses,

• for every agent a a relation a on S, called conversion,

• for every agent a a relation a on S, called preference.

From these relations we are going to define a relation called change of mind.

Definition 1 (Game) A game is a 4-uple A, S, ( a ) a∈A , ( a ) a∈A .

Example 1 (Square game 1rst version) For the first version of the square game we have:

• A = {Alice, Beth},
• S = {1|2, 1|3, 1|4, 2|3, 2|4, 2|1, 3|4, 3|1, 3|2, 4|1, 4|2, 4|3},

• Conversions Alice and Beth are given by Figure 1 • Alice = Beth and this relation is given by Figure 1 right.

Singleton equilibrium

Let us look at a first kind of equilibria.

Definition 2 (Singleton equilibrium) A singleton equilibrium is a synopsis s such that:

∀a ∈ A, s ∈ S . s a s ⇒ ¬(s a s ).
In the previous paragraphs, we have seen examples of singleton equilibria. If we are at such an equilibrium, this is fine, but if not, we may wonder how to reach an equilibrium. If s is not an equilibrium, this means that s fulfills ∃s ∈ S . s a s ∧ s a s which is the negation of ∀s ∈ S . s a s ⇒ ¬(s a s ).

The relation s a s ∧ s a s between s and s is a derived one. Let us call it change of mind for a and write it → a . We say that a changes her mind, because she is not happy with s and hopes that following → a she will reach not necessary the equilibrium, but at least a better situation. Actually since we want to make everyone happy, we have to progress along all the → a 's. Thus we consider a more general relation which we call just change of mind and which is the union of the → a 's, i.e., → a∈A → a . Now suppose that we progress along →. What happens if we reach an s from which we cannot progress further? This means ∀a ∈ A, s ∈ S . ¬(s a s ∧ s a s ) in other words, s is an equilibrium. Hence to reach an equilibrium, we progress along → until we are stuck. In graph theory, a vertex from which there is no outgoing arrow is called a sink. In relation theory it is called a minimal element;

1 1 d d d d • c c d d d d
Thus we look for sinks in the graph.

Dynamic equilibrium

Actually this progression along → is not the panacea to reach an equilibrium. Indeed it could be the case that this progression never ends, since we enter a perpetual move (think at the square game 2nd version, Figure 3). Actually we identify this perpetual move as a second kind of equilibrium.

4|2

q q ! | f A f f o | ! ' o o y f Q A 5 4|3 4|1 1|2 3|2 1|ω @ @ Ó v j G G Q Q V f u g 1|3 ! | f E E A f w f T F @ q q d j q | Ø " 3|1 m m A f v v ! | f w f T F @ I I d j q | Ø " 1|4 3|4 2|3 2|1 2|4 A f I I ! | f G G y f Q A 5 x x f o | ! '
Figure 7: A game with two SCC's

Strongly connected components

Here it is relevant to give some concepts of graph theory. A graph6 is strongly connected, if given two nodes n 1 and n 2 there is always a path going from n 1 to n 2 and a path going from n 2 to n 1 . Obviously not all the graphs are strongly connected, but they may contain some maximal subgraphs that are strongly connected; "maximal" means that one cannot add nodes without breaking the strong connectedness. Such a strongly connected subgraph is called a strongly connected components, SCC in short.

The graph below has six SCC's:

• # # • v v • # # • 3 3 v v • # # • % % • j j # # • v v • v v • C C • • t t • j j t t
The graph of Figure 3 has two SCC's (Figure 7).

From a graph, we can deduce a new graph, which we call the reduced graph (Baase [START_REF] Baase | Computer Algorithms: Introduction to Design and Analysis[END_REF] calls it condensation), whose nodes are the SCC's and the arcs are given as follows: there is an arc from an SCC S 1 to an SCC S 2 , if there exists a node n 1 in S 1 , a node s 2 in S 2 and an arc between n 1 and n 2 . By construction the reduced graph has no cycle and its strongly connected components are singletons. The reduced graphs associated with the graphs given above are as follows:

• ) ) • q q • 5 5 • q q ) ) • • {1|ω} G G {1|3, 1|2, 1|4, ...}

Dynamic equilibria as strongly connected components

At the price of extending the notion of equilibrium, we can prove that there is always an equilibrium. Indeed given a graph, we compute its reduced graph.

Then in this reduced graph, we look for sinks. There is always such a sink since in an acyclic graph (the reduced graph is always acyclic) there exists always at least a sink.

CP Equilibria are sinks in the reduced graph.

We can now split equilibria into two categories? 1. CP Equilibria (i.e., Dynamic equilibria are equilibria associated with an SCC that contains more that one synopsis.

2. Abstract Nash Equilibria (i.e., Singleton equilibria) are equilibria associated with an SCC that contains exactly one synopsis, i.e., associated with an SCC which is a singleton, hence the name singleton equilibrium.

Gene regulation networks as C/P games

In the λ phage, levels 0, 1, 2, for a gene, correspond to levels of activation or levels of concentration of the corresponding protein. Thus cI has three levels. 0 corresponds to the gene being inactive (the protein is absent), 1 corresponds to the gene being moderately active (the protein is present but moderately concentrated), 2 corresponds to the gene being highly active (the protein is concentrated). On the other hand, cro has two levels of activation, corresponding to the gene being inactive or active. Env has only one level, it is always active. A gene can move from one level at a time, as translated by the conversion relation on page 11. It has been shown that cI is a repressor for cro and a repressor for itself at level 2 and an activator for itself at level 1. This leads to the preference for cI on the left of diagram on page 12. On the other hand, it has been shown that cro is a repressor for cI, this leads to the preference and an activator for itself at level 1. This leads to the preference ⊃ for cro on the right of diagram on page 12. Moreover when both genes are inactive, the environment may lead to activate either cI or cro, this leads to the preference of the diagram on page 13.

The two equilibria correspond to two well-known states of the λ phage: the lyse and the lysogen, which the phage always reaches. In particular the lysogen { cI 2 , cro 0 , cI 1 , cro 0 } is a relatively stable state, where the phage seems inactive (dormant state). This is due to the fact that the concentration of the protein associated to cI is controlled: if it is too concentrated, a repression process makes the concentration to decrease and vice-versa if the concentration is too low an activation process makes it to increase. These antagonistic actions maintain the concentration at an intermediate level and the associated state is stable. The state cI 0 , cro 1 corresponds to what is called the lyse of the λ phage.

What is amazing in the presentation as C/P games is that these states are actually computed as C/P equilibria.

6 Conversion or preference, how to choose?

The attentive reader may have noticed that what counts is the change of mind and that there is some freedom on the conversion and the preference provided one keeps the same change of mind. More precisely, we have

→ a = a ∩ a = ( a ∪ R) ∩ a when R ∩ a = ∅ = a ∩ ( a ∪ T ) when T ∩ a = ∅
On another hand, one notices that in some examples, the preference is independent of the agent whereas in others, the conversion is independent of the agent. It seems that this is correlated with the domain of application. In particular, we may emit the following hypothesis. In biology, conversion is physics and chemistry, whereas preference is the result of evolution, then we may induce that change of mind (combination of physics and evolution) is life. Indeed since physics and chemistry is the same for everyone, it makes sense to say that conversion is the same for everybody, whereas due to evolution preference, changes with agents.

3 . 1 A

 31 simple game on a square . . . . . . . . . . . . . . . . . . . . 3.1.1 A first version . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 A second version . . . . . . . . . . . . . . . . . . . . . 3.1.3 A third version . . . . . . . . . . . . . . . . . . . . . . 3.2 Strategic games . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Prisoner's Dilemma . . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: Conversion and preference for the square game

Figure 2 :

 2 Figure 2: Agent changes of mind (on the left) and (general) change of mind (on the right) for the square game

Figure 3 :

 3 Figure 3: Conversion and change of mind for third version of the square game

Figure 4 :

 4 Figure 4: Conversion and preference in the prisoner's dilemma

Figure 5 :

 5 Figure 5: Agent and (general) change of mind in the prisoner's dilemma

Figure 6 :

 6 Figure 6: Conversion, preference and (general) change of mind in Matching Pennies

  left, with C Q for Alice and C Q for Beth ,

Figure 8 :

 8 Figure 8: Conversion, preference and change of mind for the second version of the square game

We do not draw the preference that would become to entangled.

See Section 1.1.2 of[START_REF] Osborne | An Introduction to Game Theory[END_REF] for a discussion.

In the asynchronous version.

Note the difference with the square game where players had different conversions and the same preference. The fact that the conversion is the same for everybody seems to be a feature of biologic game.

In this tutorial, when we say "graph" , we mean always "oriented graph" or "digraph".