Victor Quintero
email: victor.quintero-florez@inria.fr

Samir M Perlaza
email: samir.perlaza@inria.fr

Iñaki Esnaola
email: esnaola@sheffield.ac.uk

Jean-Marie Gorce
email: jean-marie.gorce@inria.fr

When Does Channel-Output Feedback Enlarge the Capacity Region of the Two-User Linear Deterministic Interference Channel?

Keywords: linear deterministic interference channel, noisy channel-output feedback, capacity region

The two-user linear deterministic interference channel (LD-IC) with noisy channel-output feedback is fully described by six parameters that correspond to the number of bit-pipes between each transmitter and its corresponding intended receiver, i.e., -→ n 11 and -→ n 22; between each transmitter and its corresponding non-intended receiver i.e., n12 and n21; and between each receiver and its corresponding transmitter, i.e., ←n 11 and ←n 22. An LD-IC without feedback corresponds to the case in which ← -

The capacity region of the linear deterministic interference channel with noisy channeloutput feedback is denoted by C( -→ n 11, -→ n 22, n12, n21, ←n 11, ←n 22). This paper presents the exact conditions on ←n 11 (resp. ←n 22) for observing an improvement in the capacity region C( -→ n 11, -→ n 22, n12, n21, ←n 11, 0) (resp. C( -→ n 11, -→ n 22, n12, n21, 0, ←n 22)) when compared to C( -→ n 11, -→ n 22, n12, n21, 0, 0), for any 4-tuple ( -→ n 11, -→ n 22, n12, n21) ∈ N 4 . Specifically, it is shown ∈ N 4 are also presented in this paper.

that there exists a threshold for the number of bit-pipes in the feedback link of transmitter-receiver pair 1 (resp. -→ n 22, n12, n21, 0, 0) ⊂ C( -→ n 11, -→ n 22, n12, n21, 0, ←n 22)). The exact conditions on ←n 11 resp. ←n 22 to observe an improvement on a single rate or the sum-rate capacity, for any 4-tuple ( -→ n 11, -→ n 22, n12, n21)

Introduction

Channel-output feedback is an interference management technique that significantly increases the number of degrees of freedom (DoF) for the two-user Gaussian interference channel (IC) in most of the interference regimes [START_REF] Suh | Feedback Capacity of the Gaussian Interference Channel to within 2 Bits[END_REF]. Essentially, in the very strong interference regime, the DoF gain provided by perfectchannel output feedback can be arbitrarily large when the interference to noise ratios (INRs) and signal to noise ratios (SNRs) grow to infinity. One of the reasons why feedback provides such a surprising benefit stems from the fact that it uses interference to create alternative paths to the existing point-to-point paths.

For instance, in the two-user IC, feedback creates a path from transmitter 1 (resp. transmitter 2) to receiver 1 (resp. receiver 2) in which symbols that are received at receiver 2 (resp. receiver 1) are fed back to transmitter 2 (resp. transmitter 1), which decodes the messages and retransmits them to receiver 1 (resp. receiver 2). Another metric to determine the benefits of feedback is the number of generalized DoF (GDoF), see [START_REF] Jafar | Interference Alignment: A New Look at Signal Dimensions in a Communication Network[END_REF] for a discussion on DoF and GDoF, as well as other metrics. The GDoF gain due to feedback in the IC depends on the topology of the network and the number of transmitter-receiver pairs in the network. For instance, in the symmetric K-user cyclic Z-interference channel, the DoF gain does not increase with K [START_REF] Tandon | On the Symmetric Feedback Capacity of the K-User Cyclic Z-Interference Channel[END_REF]. In particular, in the very strong interference regime, the DoF gain is shown to be monotonically decreasing with K. Alternatively, in the fully connected symmetric K-user IC with perfect feedback, the number of GDoF per user is shown to be identical to the one in the two-user case, with an exception in a particular singularity, and totally independent of the exact number of transmitter-receiver pairs [START_REF] Mohajer | On the Feedback Capacity of the Fully Connected K-User Interference Channel[END_REF]. It is important to highlight that the network topology, the number of transmitter-receiver pairs and the interference regime are not the only parameters determining the effect of feedback. Indeed, the presence of noise in the feedback links turns out to be another relevant factor. As shown later in this paper, in the case in which one transmitter-receiver pair is in a high interference regime (the interfering signal is stronger than the intended signal) and the other is in a low interference regime (the interfering signal is weaker than the intended signal), the use of feedback in the former does not enlarge the capacity region, even in the case of perfect output feedback. Conversely, using feedback in the latter might enlarge the capacity region depending on the SNR of the feedback link. The exact values of the feedback SNRs beyond which the capacity region is enlarged depend on all the other channel parameters: two forward SNRs and two forward INRs. In [START_REF] Le | Approximate Capacity Region for the Symmetric Gaussian Interference Channel with Noisy Feedback[END_REF], the capacity region of the two-user Gaussian IC (GIC) with noisy channel output feedback is approximated to within a constant number of bits for the symmetric case. These results are generalized in [START_REF] Quintero | Approximated Capacity of the Two-User Gaussian Interference Channel with Noisy Channel-Output Feedback[END_REF] for non-symmetric cases. However, from the available descriptions of the capacity regions with and without feedback, identifying whether or not the existence of a feedback link with a given SNR enlarges the capacity region is not a trivial task.

An alternative for dealing with the challenges described above is to study the GIC via its linear deterministic IC (LD-IC) approximation [START_REF] Avestimehr | Wireless Network Information Flow: a Deterministic Approach[END_REF], for which the capacity region is perfectly known [START_REF] Quintero | Noisy Channel-Output Feedback Capacity of the Linear Deterministic Interference Channel[END_REF][START_REF] Quintero | Noisy Channel-Output Feedback Capacity of the Linear Deterministic Interference Channel[END_REF]. The two-user LD-IC with noisy channel output feedback (LD-IC-NOF) is fully described by six parameters:

( - → n 11 , - → n 22 , n 12 , n 21 , ← - n 11 , ← - n 22 ) ∈ N 6 .
There exists a mapping between the parameters describing the two-user LD-IC and the parameters describing the GIC. More specifically, there are two forward SNRs ( --→ SNR i 1); two forward INRs (INR ij 1); and two backward SNRs ( ←--SNR i 1), with i ∈ {1, 2} and j ∈ {1, 2} \ {i}. In the LD-IC, the parameters of the GIC are mapped into the number of bit-pipes between each transmitter and its corresponding intended receiver, i.e., -→ n ii = 1 2 log 2 ( --→ SNR i ) ; between transmitter j and receiver i i.e., n ij = 1 2 log 2 (INR ij ) ; and between each receiver and its corresponding transmitter, i.e., ← - Based on these results, several relevant engineering questions arise in this setting. For instance, in which of the two transmitter-receiver pairs must the feedback link be implemented if the objective is to improve: (a) the individual rate of the transmitter-receiver pair in which feedback is implemented; (b) the individual rate of the other transmitter-receiver pair; or (c) the sum-rate of both transmitter-receiver pairs. In each of these scenarios, the feedback SNR, either ←n 11 or ←n 22 , must be bigger than a given threshold for the improvement to be observed. Interestingly, the answer is completely different for each of these scenarios. In addition to the results described above, the exact values of ←n 11 or ←n 22 for which feedback does not enlarge the capacity region are also identified.

n ii = 1 2 log 2 ( ←-- SNR i ) .

Linear Deterministic Interference Channel with Noisy-Channel Output Feedback

Consider the two-user LD-IC-NOF, with parameters -→ n 11 , -→ n 22 , n 12 , n 21 , ←n 11 and ←n 22 described in Fig. 1. The parameters -→ n ii , n ij and ←n ii with i ∈ {1, 2} and j ∈ {1, 2} \ {i}, are non-negative integers. Parameter -→ n ii represents the number of bit-pipes between transmitter i and receiver i; parameter n ij represents the number of bit-pipes between transmitter j and receiver i; and parameter ←n ii represents the number of bit-pipes between receiver i and transmitter i (feedback). At transmitter i, with i ∈ {1, 2}, the channel-input

X (n) i during channel use n, with n ∈ {1, . . . , N }, is a q-dimensional binary vector X (n) i = Ä X (n) i,1 , . . . , X (n) i,q ä T , with q = max ( - → n 11 , - → n 22 , n 12 , n 21 ) , (1)
T X 1 T X 2 RX 2 RX 1 n 11 n 22 X 1,1 X 1,5 X 1,1 ! n 11 ! n 22 n 12 n 21

Signal

Interference Feedback and N the block length. At receiver i, the channel-output

- → Y (n) i during channel use n is also a q-dimensional binary vector - → Y (n) i = Ä - → Y (n) i,1 , . . . , - → Y (n) i,q
ä T . Let S be a q × q lower shift matrix. The input-output relation during channel use n is given as follows

- → Y (n) i =S q-- → n ii X (n) i + S q-nij X (n) j , (2)
and the feedback signal ← -Y i,n available at transmitter i at the end of channel use n satisfies

Ç (0, . . . , 0) , Å ← - Y (n) i ã T å T =S (max( - → n ii,nij )-← - n ii) + -→ Y (n-d) i , (3)
where d is a finite feedback delay, additions and multiplications are defined over the binary field, and (•) + is the positive part operator. The dimension of the vector (0, . . . , 0) in

(3) is q -min ← - n ii , max( - → n ii , n ij ) and the vector ← - Y (n) i represents the min ← - n ii , max( - → n ii , n ij ) least significant bits of S (max( - → n ii,nij )-← - n ii) + -→ Y (n-d) i . Transmitter i sends M i information bits b i,1 , . . . , b i,Mi by sending the code- word Ä X (1) i , . . . , X (N ) i ä .
The encoder of transmitter i can be modeled as a set of deterministic mappings f

(1) i , . . . , f (N ) i , with f (1) i : {0, 1} Mi → {0, 1} q and ∀n ∈ {2, . . . , N }, f (n) i : {0, 1} Mi × {0, 1} q(n-1) → {0, 1} q , such that X (1) i =f (1) i b i,1 , . . . , b i,Mi and (4)
X (n) i =f (n) i b i,1 , . . . , b i,Mi , ← - Y (1) i , . . . , ← - Y (n-1) i . (5)
At the end of the block, receiver i uses the sequence Y

i , . . . , Y (N ) i to generate the estimates bi,1 , . . . , bi,Mi . The average bit error probability at receiver i, denoted by p i , is calculated as follows

p i = 1 M i Mi =1 1 { bi, =b i, } . (6)
A rate pair (R 1 , R 2 ) ∈ R 2 + is said to be achievable if it satisfies the following definition.

Definition 1 (Achievable Rate Pairs). The rate pair (R 1 , R 2 ) ∈ R 2
+ is achievable if there exists at least one pair of codebooks X N 1 and X N 2 with codewords of length N , and the corresponding encoding functions f

(1) 1 , . . . , f (N ) 1 and f (1) 2 , . . . , f (N ) 2
such that the average bit error probability can be made arbitrarily small by letting the block length N grow to infinity.

The information capacity region is the closure of all information achievable rate pairs (R ) of the two-user LD-IC-NOF is the set of non-negative rate pairs (R 1 , R 2 ) that satisfy ∀i ∈ {1, 2} and j ∈ {1, 2} \ {i}:

R i min (max ( - → n ii , n ji ) , max ( - → n ii , n ij )) , (7a)
R i min Ä max ( - → n ii , n ji ) , max Ä - → n ii , ← - n jj -( - → n jj -n ji ) + ää , (7b)
R 1 +R 2 min max ( - → n 22 , n 12 ) + ( - → n 11 -n 12 ) + , max ( - → n 11 , n 21 ) + ( - → n 22 -n 21 ) + , (7c)
R 1 +R 2 max ( - → n 11 -n 12 ) + , n 21 , - → n 11 -(max ( - → n 11 , n 12 ) -← - n 11 ) + + max ( - → n 22 -n 21 ) + , n 12 , - → n 22 -(max ( - → n 22 , n 21 ) -← - n 22 ) + , (7d) 2R i +R j max ( - → n ii , n ji ) + ( - → n ii -n ij ) + + max ( - → n jj -n ji ) + , n ij , - → n jj -(max ( - → n jj , n ji ) -← - n jj ) + . ( 7e
)
3 Preliminaries

Definitions

Let α i ∈ Q be the interference regime of transmitter-receiver pair i, with i ∈ {1, 2} and j ∈ {1, 2} \ {i},

α i = n ij - → n ii . (8)
For each transmitter-receiver pair i, there exist five possible interference regimes (IRs): very weak IR (VWIR), i.e., α i 1 2 , weak IR (WIR), i.e., 1 2 < α i 2 3 , moderate IR (MIR), i.e., 2 3 < α i < 1, strong IR (SIR), i.e., 1 α i 2 and very strong IR (VSIR), i.e., α i > 2 [START_REF] Etkin | Gaussian Interference Channel Capacity to within One Bit[END_REF]. The scenarios in which the desired signal is stronger than the interference (α i < 1), namely VWIR, WIR and MIR, are referred to as low-interference regimes (LIRs). Conversely, the scenarios in which the desired signal is weaker than or equal to the interference (α i 1), namely SIR and VSIR, are referred to as high-interference regimes (HIRs). In the twouser LD-IC, up to twenty-five possible interference regimes might emerge, given α 1 and α 2 . However, only eight cases are of real interest. This is because the transmitter-receiver pairs can be indifferently labeled and thus, for instance, studying the case in which α 1 1 2 and α 2 > 2 is the same as studying the case in which α 1 > 2 and α 2 1 2 .

The main results of this paper are presented using a list of events (Boolean variables) that are fully determined by the parameters -→ n 11 , -→ n 22 , n 12 , and n 21 .

For instance, given the parameters ( -→ n 11 , -→ n 22 , n 12 , n 21 ), the events ( 9)-(??) describe some combinations of interference regimes that are particularly interesting. Let i ∈ {1, 2} and j ∈ {1, 2} \ {i} and define the following events:

E 1 : α 1 <1 ∧ α 2 <1, (i and j in LIR) (9)
E 2,i :

α i 1 2 ∧ 1 α j 2, (i in VWIR and j in SIR) (10)
Note that the event E 6,i includes α i = 1 and excludes α j = 1, and the event E 7,i includes α j = 1. Some other auxiliary events are considered. The event in which the signal from transmitter i is stronger in its intended receiver than in its non-intended receiver is denoted by E 8,i , i.e.,

E 8,i : - → n ii >n ji . (16)
The event in which the sum of the number of bit-pipes in the direct links is bigger than the sum of the number of bit-pipes in the cross-interference links is denoted by E 9 , i.e.,

E 9 : - → n 11 + - → n 22 >n 12 + n 21 . (17)
The event in which the sum of the number of bit-pipes in the direct links is bigger than the sum of the number of bit-pipes in one cross-interference link and twice the number of the bit-pipes in the other cross-interference link is denoted by E 10,i , i.e.,

E 10,i : - → n ii + - → n jj >n ij + 2n ji . (18)
Finally, the event in which the sum of the number of bit-pipes in the direct links is bigger than the number of bit-pipes in one cross-interference link is denoted by E 11,i , i.e.,

E 11,i : - → n ii + - → n jj <n ij . (19)
In the following, given an event, e.g.

E 8,i : - → n ii > n ji , the notation ‹ E 8,i implies - → n ii < n ji ; E 8,i implies - → n ii n ji ; and Ě8,i implies - → n ii n ji .
Combining the events ( 9)-( 19), five main events are identified:

S 1,i : (E 1 ∧ E 8,i )∨(E 2,i ∧ E 8,i )∨(E 3,i ∧ E 8,i ∧ E 9 )∨(E 4,i ∧ E 8,i ∧ E 9 )∨(E 5,i ∧ E 8,i ∧ E 9 ) , (20)
S 2,i : Ä E 3,i ∧ ‹ E 8,j ∧ E 9 ä ∨ Ä E 6,i ∧ ‹ E 8,j ∧ E 9 ä ∨ Ä ‹ E 1 ∧ ‹ E 8,j ä , (21)
S 3,i : E 1 ∧ E 8,i ∨ E 2,i ∧ E 8,i ∨ E 3,i ∧ Ě8,j ∧ E 8,i ∨ E 4,i ∧ Ě8,j ∧ E 8,i ∨ E 5,i ∧ Ě8,j ∧ E 8,i ∨ E 1 ∧ Ě8,j ∨ (E 7,i ) , (22) S 4 : E 1 ∧ E 8,1 ∧ E 8,2 ∧ E 10,1 ∧ E 10,2 , (23)
S 5 : E 1 ∧ E 11,1 ∧ E 11,2 . (24)
For all i ∈ {1, 2} the events S 1,i , S 2,i and S 3,i exhibit the property stated by the following lemma.

Lemma 2. For all i ∈ {1, 2} and for all ( -→ n 11 , -→ n 22 , n 12 , n 21 ) ∈ N 4 , only one of the events among S 1,i , S 2,i and S 3,i is true.

Proof: The proof follows from verifying that for all i ∈ {1, 2} and j ∈ {1, 2} \ {i}, the events (20)-( 22) are mutually exclusive. For instance, consider that the event (E 1 ∧ E 13,i ) in (20) is true. Then, S 1,i is true and E 2,i , E 3,i , E 4,i , E 5,i , E 6,i , E 7,i and E 8,i hold false, which implies that S 2,i and S 3,i hold false as well, since all events in (21) and ( 22) are false. The same verification can be made for all the remaining events in (20). This proves that if S 1,i is true then S 2,i and S 3,i hold simultaneously false. The same verification can be done for showing that when S 2,i holds true (resp. S 3,i ), both events S 1,i and S 3,i (resp. S 1,i and S 2,i ) hold simultaneously false. Finally following the same reasoning it can be verified that if any pair of the events {S 1,i , S 2,i , S 3,i } is false, the remaining event is necessarily true. This completes the proof.

The events S 4 and S 5 exhibit the property stated by the following lemma.

Lemma 3. if one of the events S 4 or S 5 holds true, then the other necessarily holds false.

Proof: The proof of Lemma 3 follows along the same lines of the proof of Lemma 2.

Rate Improvement Metrics

The rate improvements are given in terms of the following metrics [START_REF] Quintero | Noisy Channel-Output Feedback Capacity of the Linear Deterministic Interference Channel[END_REF][START_REF] Quintero | Noisy Channel-Output Feedback Capacity of the Linear Deterministic Interference Channel[END_REF]: (a) maximum individual rate improvements ∆ 1 and ∆ 2 ; and (b) maximum sum-rate improvement Σ, with ∆ i ∈ R + and Σ ∈ R + for i ∈ {1, 2}.

Let ), due to the effect of channel-output feedback with respect to the case without feedback is

C 1 = C( - →
∆ i ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n 11 , ← - n 22 ) =max Rj >0 sup (Ri,Rj )∈C1 R i - sup (R † i ,Rj )∈C2 R † i , (25)
and the maximum improvement of the sum rate Σ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n 11 , ←n 22 ) with respect to the case without feedback is

Σ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n 11 , ← - n 22 ) = sup (R1,R2)∈C1 R 1 + R 2 - sup (R † 1 ,R † 2 )∈C2 R † 1 + R † 2 . (26)
In the following, when feedback is exclusively used by transmitter-receiver pair i, i.e., ←n ii > 0 and ←n jj = 0, then the maximum improvement of the individual rate of transmitter-receiver k, with k ∈ {1, 2}, and the maximum improvement of the sum rate are denoted by 4 Main Results

∆ k ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n ii ) and Σ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n ii ), respectively. Hence, this notation ∆ k ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n ii ) replaces either ∆ k ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n 11 , 0) or ∆ k ( - → n 11 , - → n 22 , n 12 ,

Enlargement of the Capacity Region

In this subsection, the capacity region of a two-user LD-IC-NOF with parameters ( -→ n 11 , -→ n 22 , n 12 , n 21 ), when feedback is available only at transmitter-receiver pair i, i.e., ←n ii > 0 and ← -

n jj = 0, is denoted by C ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n ii ) instead of C ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n 11 , 0) or C ( - → n 11 , - → n 22 , n 12 , n 21 , 0, ← - n 22
), when i = 1 or i = 2, respectively. Following this notation, Theorem 1 identifies the exact values of ← - Proof: The proof of Theorem 1 is presented in [START_REF] Quintero | When Does Channel-Output Feedback Enlarge the Capacity Region of the Two-User Linear Deterministic Interference Channel?[END_REF]. Theorem 1 shows that under event S 3,i in (22), implementing feedback in transmitter-receiver pair i does not bring any capacity region enlargement. Alternatively, under events S 1,i in (20) and S 2,i in (21), the capacity region can be enlarged when ← -

n ii for which the strict inclusion C ( - → n 11 , - → n 22 , n 12 , n 21 , 0, 0) ⊂ C ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n ii ) holds, with i ∈ {1, 2}. Theorem 1. Let i ∈ {1, 2}, j ∈ {1, 2} \ {i} and ← - n * ii ∈ N be ← - n * ii = ® max Ä n ji , ( - → n ii -n ij ) + ä if S 1,i = True - → n jj + ( - → n ii -n ij ) + if S 2,i = True. ( 27
n ii > ← - n * ii .
That is, there exists a threshold on the SNR of the feedback link beyond which it is possible to observe a capacity region enlargement.

An interesting observation is that the threshold ←n * ii beyond which feedback is useful is different under event S 1,i in (20) and event S 2,i in (21). In general when S 1,i holds true, the enlargement of the capacity region is due to the fact that feedback allows using interference as side information [START_REF] Shannon | Channels with side information at the transmitter[END_REF]. More specifically, when feedback is used at transmitter-receiver pair i and ← -

n ii > max Ä n ji , ( - → n ii -n ij ) + ä
, transmitter i obtains part of the information sent by transmitter j. This information can be re-transmitted by transmitter i to cancel the interference it produced at receiver i when it was first transmitted by transmitter j. Interestingly, the interference perceived at receiver j due to this re-transmission can be cancelled given that this information was reliably decoded when it was first sent by transmitter j. This allows transmitter-receiver pair i or j to improve its individual rate.

Alternatively, when S 2,i in (21) holds true, the enlargement of the capacity region occurs thanks to the fact that some of the bits that cannot be transmitted directly from transmitter j to receiver j, that is, those transmitted via the bitpipes -→ n jj + 1, . . . , max ( -→ n jj , n ij ), can arrive to receiver j via an alternative path: transmitter j -receiver i -transmitter i -receiver j. For this to be possible at least the

Ä - → n jj + ( - → n ii -n ij ) + + 1 ä -th (
n jj = 0 in the two- user LD-IC-NOF. Theorem 2. Let i ∈ {1, 2}, j ∈ {1, 2} \ {i} and ← - n † ii ∈ N be ← - n † ii = max Ä n ji , ( - → n ii -n ij ) + ä . ( 28
)
Assume that either S 2,i = True or S 3,i = True. Then, for all ← -

n ii ∈ N, ∆ i ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n ii ) = 0. Assume that S 1,i = True. Then, when ← - n ii ← - n † ii , it holds that ∆ i ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n ii ) = 0; and when ← - n ii > ← - n † ii , it holds that ∆ i ( - → n 11 , - → n 22 , n 12 , n 21 , ← - n ii ) > 0.
rates. Under event S 2,i , using feedback in transmitter-receiver pair i, with ← -

n ii > ← - n ‡
ii , exclusively benefits transmitter-receiver pair j, which can improve its own individual rate.

Improvement of the Sum-Capacity

Implementing channel output feedback in transmitter-receiver pair i increases the sum-capacity under certain conditions. Theorem 4 identifies the exact values of ←n ii for which the sum-capacity can be improved, for parameters ( -→ n 11 , -→ n 22 , n 12 , n 21 ) and ←n jj = 0 in the two-user LD-IC-NOF. Proof: The proof of Theorem 4 is presented in [START_REF] Quintero | When Does Channel-Output Feedback Enlarge the Capacity Region of the Two-User Linear Deterministic Interference Channel?[END_REF]. Theorem 4 identifies the conditions under which implementing feedback in transmitter-receiver pair i improves the sum-capacity whenever ←n ii > ←n + ii , that is, Σ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ) > 0. Theorem 4 highlights that one of the necessary but not sufficient conditions for improving the sum-capacity by implementing feedback in transmitter-receiver pair i is that either (a) both transmitterreceiver pairs must be in LIR; or (b) both transmitter-receiver pairs must be in HIR. This follows immediately from observing that for S 4 or S 5 to hold true, the event E 1 or E 1 must hold true.

Theorem 4. Let i ∈ {1, 2}, j ∈ {1, 2} \ {i} and ← - n + ii ∈ N be ← - n + ii = ® max Ä n ji , ( - → n ii -n ij ) + ä if S 4 = True - → n jj + ( - → n ii -n ij ) + if S 5 =
Interestingly, Theorem 4 shows that if one transmitter-receiver pair is in LIR and the other transmitter-receiver pair is in HIR, then the sum-capacity cannot be improved. Finally, note that the thresholds ←n + ii in the events S 4 and S 5 coincide with those observed in Theorem 1.

Conclusions

This paper presented the exact conditions on the feedback parameters ←n 11 and ←n 22 , beyond which the capacity region of the two-user LD-IC-NOF can be enlarged for any [START_REF] Tandon | On the Symmetric Feedback Capacity of the K-User Cyclic Z-Interference Channel[END_REF]

  region, i.e., C( -→ n 11, -→ n 22, n12, n21, 0, 0) ⊂ C( -→ n 11, -→ n 22, n12, n21, ←n 11, 0) (resp. C( -→ n 11,

Fig. 1 .

 1 Fig. 1. Two-user linear deterministic interference channel with noisy channel-output feedback. The bit-pipe line number 1 represents the most significant bit.

n

  21 , 0, ←n 22 ), when i = 1 or i = 2, respectively. The same holds for the notation Σ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ) that replaces Σ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n 11 , 0) or Σ( -→ n 11 , -→ n 22 , n 12 , n 21 , 0, ←n 22 ), when i = 1 or i = 2, respectively.

  n 11 , -→ n 22 , n 12 , n 21 , ←n 11 , ←n 22 ) and C 2 = C( -→ n 11 , -→ n 22 , n 12 , n 21 , 0, 0) be the capacity region with noisy channel-output feedback and without feedback, respectively. In order to formally define ∆ 1 , ∆ 2 and Σ, consider a two-user LD-IC-NOF with parameters -→ n 11 , -→ n 22 , n 12 , n 21 , ←n 11 , and ←n 22 . The maximum improvement of the individual rate R i , ∆ i ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n 11 , ←n 22

)

  Assume that S 3,i = True. Then, for all ←n ii ∈ N, C ( -→ n 11 , -→ n 22 , n 12 , n 21 , 0, 0) = C ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ). Assume that either S 1,i = True or S 2,i = True. Then, for all ←n ii ←n * ii , C ( -→ n 11 , -→ n 22 , n 12 , n 21 , 0, 0) = C( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ) and for all ←n ii > ←n * ii , C ( -→ n 11 , -→ n 22 , n 12 , n 21 , 0, 0) ⊂ C ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ).

  True. (30) Assume that S 4 = False and S 5 = False. Then, for all ←n ii ∈ N, Σ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ) = 0. Assume that S 4 = True or S 5 = True. Then, when ←n ii ←n + ii , it holds that Σ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ) = 0; and when ←n ii > ←n + ii , it holds that Σ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ) > 0.

  -tuple ( -→ n 11 , -→ n 22 , n 12 , n 21 ) ∈ N 4 . More specifically, the exact values of ←n 11 (resp. ←n 22 ) for which C( -→ n 11 , -→ n 22 , n 12 , n 21 , 0, 0) ⊂ C( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n 11 , 0) (resp. C( -→ n 11 , -→ n 22 , n 12 , n 21 , 0, 0) ⊂ C( -→ n 11 , -→ n 22 , n 12 , n 21 , 0, ←n 22 )), with strict inclusion. The exact conditions on ←n 11 (resp. ←n 22 ) to observe an improvement on a single rate or the sum-rate capacity, for any 4-tuple ( -→ n 11 , -→ n 22 , n 12 , n 21 ) ∈ N 4 were also presented. Interestingly, there exist conditions in the two-user LD-IC-NOF in which the use of feedback does not enlarge the capacity region.

X 1,2 X 1,3 X 1,4 X 1,5 X 2,4 X 2,3 X 2,2 X 2,1 X 1,1 X 1,2 X 1,3 L X 2,1 X 1,4 L X 2,2 X 1,5 L X 2,3 X 2,1 L X 1,2 X 2,2 L X 1,3 X 2,3 L X 1,4X 2,4 L

The CITI Laboratory is a joint lab between the This research was supported in part by the European Commission under Marie Sklodowska-Curie Individual Fellowship No. 659316 , and the Administrative Department of Science, Technology and Innovation of Colombia (Colciencias), fellowship No. 617-2013.

Proof: The proof of Theorem 2 is presented in [START_REF] Quintero | When Does Channel-Output Feedback Enlarge the Capacity Region of the Two-User Linear Deterministic Interference Channel?[END_REF]. Theorem 2 highlights that under events S 2,i in (21) and S 3,i in (22), the individual rate R i cannot be improved by using feedback in transmitter-receiver pair i, i.e., ∆ i ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ) = 0. Alternatively, under event S 1,i in (20), the individual rate R i can be improved, i.e.,

. It is worth noting that under event S 3,i in (22), the capacity region cannot be improved via feedback (Theorem 1) and thus, none of the individual rates can be improved as suggested by Theorem 2. Alternatively, under event S 2,i in (21), the capacity region can be enlarged (Theorem 1) but the individual rate R i cannot be improved (Theorem 2). This implies that the capacity improvement occurs due to the fact that R j can be improved. More specifically, in this case:

This implies that using feedback in transmitter-receiver pair i is exclusively beneficial for transmitter-receiver pair j, as shown in the following section.

Improvement of the Individual Rate R j by Using Feedback in Link i

Implementing channel output feedback in transmitter-receiver pair i might allow increasing the individual rate of transmitter-receiver pair i or j; or both individual rates. This reveals the altruistic nature of implementing feedback as suggested in [START_REF] Perlaza | Perfect Output Feedback in the Two-User Decentralized Interference Channel[END_REF]. Theorem 3 identifies the exact values of ←n ii for which the individual rate R j can be improved by using feedback in transmitter-receiver pair i, given the parameters ( -→ n 11 , -→ n 22 , n 12 , n 21 ) and ←n jj = 0 in the two-user LD-IC-NOF.

Assume that S 3,i = True. Then, for all ←n ii ∈ N, ∆ j ( -→ n 11 , -→ n 22 , n 12 , n 21 , ←n ii ) = 0. Assume that either S 1,i = True or S 2,i = True. Then, when ← -

Proof: The proof of Theorem 3 is presented in [START_REF] Quintero | When Does Channel-Output Feedback Enlarge the Capacity Region of the Two-User Linear Deterministic Interference Channel?[END_REF]. Theorem 3 shows that under event S 3,i in (22), implementing feedback in transmitter-receiver pair i does not bring any improvement on the rate R j . This is in line with the results of Theorem 1 that states that under event S 3,i in (22), implementing feedback in transmitter-receiver pair i does not enlarge the capacity region.

In contrast, under events S 1,i in (20) and S 2,i in (21), the individual rate

It is important to highlight that under event S 1,i , the threshold on ←n ii for increasing the individual rate R i i.e., ( ←n † ii ), and R j i.e., ( ←n ‡ ii ), are identical, see Theorem 2 and Theorem 3. This shows that in this case, the use of feedback in transmitterreceiver pair i, with ← -

ii , simultaneously improves both individual


Stephane Gauthier
email: stephane.gauthier@univ-paris1.fr.

Taraneh Tabatabai
email: tara.tabatabai@gmail.com.

Thierry Coville

Santiago Garriga

Farhad

François Bourguignon

How incentives matter ? An illustration from the Targeted Subsidies reform in Iran *

Keywords: JEL classification: D12, D82, H21, L51 Principal-agent, incentive constraints, Iran, Targeted Subsidies, social valuations, AIDS

We use the Targeted Subsidies Reform implemented in Iran in 2011 to recover empirically the social valuations of Iranian households relying on the assumption of optimal taxes. Unlike the existing literature, we do not restrict attention to a specific pattern for the incentive constraints associated with nonlinear income taxation. Instead we recover the Lagrange multipliers corresponding to these constraints. We find evidence of a significant redistribution toward the bottom three deciles of the income distribution before the reform. This redistribution is however limited by an incentive constraint where the rich envy the social treatment of the poor. At the outcome of the reform incentives no longer matter and the social welfare function of the government of Iran displays a Benthamite-like form.

Introduction

Most theoretical models of public intervention into the economy rely on the principal-agent setup and so emphasize incentive considerations due to the presence of asymmetric information between the two parties. The regulator is assumed to deal with a firm better informed about its cost, and the tax authority has to design taxes observing neither labor skill nor effort of taxpayers. Asymmetric information limits the scope of public intervention. Large compensating transfers to high cost firms may lead more efficient firms to misreport their cost, and some high skilled taxpayers in the face of pro-poor income support programs may have an incentive to relax labor effort. Although incentive issues often seem theoretically sound, in practice little is known about how much they actually constraint public intervention.

The existing empirical literature is mostly aimed at detecting the presence of asymmetric information. The standard strategy relies on the hypothesis that agents differ according to a few dimension and have preferences satisfying a suitable version of the single-crossing condition. Under these assumptions only the local (downward) incentive constraints are relevant. One can then infer the presence of asymmetric information from the market outcome associated with this pattern of incentives. In insurance markets, for instance, this pattern corresponds to the situation where high risk insurees are ready to switch to the contract designed for low risk. If so, the optimal response of insurers consists to reduce the coverage of low risk. The offered contracts thus display a positive correlation between risk and coverage that reveals the presence of asymmetric information. This positive correlation test first proposed by [START_REF] Chiappori | Testing for asymmetric information in insurance markets[END_REF] yields 'decidedly mixed' evidence about asymmetric information [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF].

The pattern of local downward incentive constraints is sometimes used directly as an identifying assumption that allows us to recover some underlying parameters of the economy. For instance the estimated weight put on firms' profits by the regulatory authority of the French water industry in [START_REF] Bontemps | Identification and estimation of incentive contracts under asymmetric information: an application to the French water sector[END_REF] is derived from first-order conditions for optimal contracts that are valid if low cost firms may choose the contract designed for the less efficient firms. In the public finance literature [START_REF] Bourguignon | Tax-benefit revealed social preferences[END_REF] estimate the social valuations of various classes of taxpayers, who only differ according to labor skill, from first-order conditions for optimal income taxes assuming that conditions for the local downward neighboring incentive constraints are satisfied; see also, e.g., [START_REF] Immervoll | Welfare reform in European countries: a microsimulation analysis[END_REF], [START_REF] Eissa | Evaluation of four tax reforms in the United States: labour supply and welfare effects for single mothers[END_REF] or [START_REF] Blundell | Optimal income taxation of lone mothers: an empirical comparison for Germany and the UK[END_REF] for running similar exercises.

As stressed by [START_REF] Finkelstein | Multiple dimensions of private information: evidence from the long-term care insurance market[END_REF] the reduction of individual heterogeneity to a few dimensions may be overly restrictive in insurance markets. This reduction is also problematic when agents are consumers who choose among many different consumption goods since we know from the econometrics of demand that a high number of free parameters is needed to achieve a flexible demand form. [START_REF] Finkelstein | Testing for asymmetric information using 'unused observables' in insurance markets: evidence from the U.K. annuity market[END_REF] propose new empirical tests for detecting asymmetric information in the case where insurees differ according to several dimensions. Still, it remains unclear how the estimates of the underlying parameters that we recover depend on the additional theoretical restrictions introduced to deal with incentive constraints, e.g., single crossing conditions.

In our paper, unlike the main strand of the literature, we do not a priori restrict attention to some specific pattern of incentive constraints. Instead we rely on the assumption of an optimal public policy to recover this pattern. This is done by working with firstorder conditions that still depend on the Lagrange multipliers associated with incentive constraints. These multipliers are estimated and, when found positive, the Kuhn-Tucker conditions imply that the associated constraints are binding. This strategy allows us to identify precisely the limits to redistribution due to asymmetric information, rather than simply detect whether asymmetric information operates or not in the market.

This methodology is illustrated on the Targeted Subsidies reform recently implemented in Iran. Until 2011 Iran was using the oil wealth to finance massive consumption subsidies. President Ahmadinejad launched in December 2010 a large reform that replaced the subsidies with a VAT-like scheme accompanied with compensating income transfers (see [START_REF] Guillaume | The chronicles of the Subsidy Reform[END_REF]. [START_REF] Gahvari | Fiscal reforms in general equilibrium: theory and an application to the subsidy debate in Iran, The B[END_REF] and Gharibnavaza and Waschik (2015) discuss the redistributive impact of this reform abstracting from incentive issues. When taken into account, our results show that the reform has about equalized income class valuations. We find that incentive considerations matter before the reform, but no longer at its outcome. Before the reform, incentives only involve the rich and the poor: the redistribution toward the poor appears severely limited by the social desire to preserve a rich small elite. The middle income class households are never concerned by incentives, a feature that is inconsistent with the standard local downward pattern. In order to assess how our measures of the quantitative importance of incentives relate to incorrect assumptions about the pattern of incentive constraints, we run a counterfactual exercise where we a priori restrict attention to the familiar local downward pattern. The importance of incentives then is biased downwards.

The paper is organized as follows. Sections 2 and 3 derive necessary conditions for optimal consumption and income taxes that depend on Lagrangian multipliers associated with incentive constraints. Section 4 presents the Targeted subsidies reform and consumption data that we use. Section 5 finally gives the Lagrangian multipliers and the households' social valuations recovered from Iranian data.

Theoretical benchmark

The economy is populated with one government and a total unit size continuum of agents. There are n i agents of type i (i = 1, . . . , I) whose preferences are represented by the utility function u i (x, y), where x is a bundle of K consumption goods and y stands for beforetax income. The utility function is increasing quasi-concave in (x, -y). The (possibly multidimensional) index i is private information to the agent.

The government has to finance an amount G = 0 of public expenditures. It can use linear consumption taxes t and nonlinear income taxes. Assuming fixed producer prices p, the vector of consumer prices is q = p + t. The income tax is y -R(y) when the before-tax income is y.

Given y i and the after-tax income R i = R(y i ) a type i agent consumes ξ i (q, y i , R i ) that maximizes u i (x, y i ) subject to her budget constraint q x ≤ R i . Her indirect utility is v i (q, y i , R i ). Let (a i ) be the profile of exogenous social weights. Given (y i ) the government chooses q and (R i ) that maximize

i a i n i v i (q, y i , R i ) (1)
subject to the feasibility constraint

i n i (q -p) ξ i (q, y i , R i ) + i n i (y i -R i ) ≥ G, (2)
and the incentive constraints

v i (q, y i , R i ) ≥ v i (q, y j , R j ). (3)
for all i and j = i. Let ρ and (λ ij ) be the Lagrange multipliers associated with (2) and (3). The first-order condition for R i to solve the government's program is

n i a i α ii + ρn i t ∂ξ ii ∂R -1 + j =i (λ ij α ii -λ ji α ji ) = 0, (4)
where

α ji = ∂v j ∂R (q, y i , R i )
stands for the marginal utility of income of a type j agent endowed with (y i , R i ). Using (4) and appealing to the Slutsky decomposition,

∂ξ ii ∂q k = ∂ ξ ii ∂q k -ξ k ii ∂ξ ii ∂R ,
where ξ ij is the compensated demand of good for a type i agent endowed with the income bundle (y j , R j ), the first-order condition in q k can be written

i n i t ∂ ξ ii ∂q k + i j =i α ij λ ij ρ ξ k ij -ξ k jj = 0. (5)
Optimal consumption taxes are uniform if incentives do not matter, i.e., if λ ij is 0 for every i and j. Otherwise they should be designed to relax incentive constraints: when type i mimics type j (λ ij > 0), the consumption of good k should be discouraged when type i agents have a greater consumption of good k than the mimicked type j agents.

3 Targeted Subsidies reform

Empirical strategy

Our aim is to assess the relevant pattern of incentives and the redistributive stance over the period where the Targeted Subsidies reform was implemented in Iran. We rely on the assumption that the first-order conditions ( 4) and ( 5) for optimal taxes are satisfied. This system of I + K first-order conditions is linear in the

I + I(I -1) = I 2 ratios a * i = a i α ii ρ and λ * ij = α ij λ ij ρ .
These are the parameters that we want to recover empirically. A 1 rial transfer toward one type i agent yields a * i rials to the society, which gives a measure for type i social valuation. Similarly, if the government were able to track type i agents when they mimic type j, a 1 rial tax paid by every such type i agents would yield λ * ij rials to the society. The relevant pattern of incentives is characterized by the constraints associated with the multipliers λ * ij that are positive. This pattern delineates the limits to redistribution implied by asymmetric information about agents' types: if λ * ij > 0 for some i, a greater transfer toward type j agents is socially valuable but it cannot be implemented because it would imply a violation of incentive requirements.

It is convenient to rewrite (4) and ( 5) in a form that is better suited for empirical purpose, switching to expenditures and demand elasticities,

n i a * i = n i 1 - t val 1 + t val n i q ξ ii n i R i ε R ii - j =i α ii α ij λ * ij -λ * ji , (6)
t val 1 + t val i n i q k ξ k ii q k ξ k εk ii + i j =i λ * ij n i n i q k ξ k ij q k ξ k - n i n j n j q k ξ k jj q k ξ k = 0, (7)
with ad valorem taxes (t val ) defined by

t q = t val 1 + t val .
The income and compensated price elasticity of demand ε R ij and εk ij of good apply to a type i agent earning y j before tax and R j after tax.

Once endowed with estimates for elasticities, our data leaves us with (a * i ) and (λ * ij ) as the only unknowns in ( 6) and [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF]. Since the profile (a * i ) is absent from [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF], one can recover these unknowns by proceeding in two steps. In the first step we recover the profile of the I(I -1) multipliers (λ * ij ) by minimizing the sum of the K squared left-hand side of [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF]. Second, we reintroduce these multipliers into the I equations ( 6) to get the I social valuations (a * i ).

Data from Iran

The first-order conditions ( 6) and ( 7) involve consumption and tax data. Consumption data comes from the Household Income and Expenditure Survey (HIES) collected every year from 1388 to 1393 by the Statistical Center of Iran. The HIES provides us with household expenditures on COICOP categories of goods, as well as some households demographic variables. Our sample consists of urban literate households between the ages of 18 and 70. 1They are allocated to three income classes: the poor consist of the bottom three deciles of the per consumption unit total (positive) expenditures distribution while the rich are in the top decile of this distribution. To deal with high inflation that prevails over the sample window, we build these classes referring to each year separately, e.g., the poor income class consists of the households that belong to the bottom three deciles in 1388, plus those in the same quantiles in 1389, and so on. The HIES gives expenditures q k ξ k ii on category k and total expenditures R i of each class i household. Categories of consumption goods are aggregated according to the 2-digit level of the COICOP international classification. 2 Appendix A shows that the main differences in the consumption of the various classes relate to Food, Clothing and Health. Food appears inferior whereas the budget shares of Clothing and Health are up to three times higher for the rich than the poor.

The consumption subsidies used in Iran before the Targeted Subsidies reform are difficult to assess because of hidden components that are not reflected in the official tax scale [START_REF] Esfahani | Hidden public expenditures and the economy in Iran[END_REF]. In what follows we impute the subsidies used by [START_REF] Gahvari | Fiscal reforms in general equilibrium: theory and an application to the subsidy debate in Iran, The B[END_REF] to year 1388 of the Iranian calendar (from March 21, 2009), which constitutes our pre-reform period. Appendix A shows that the subsidies are concentrated on Transport and Food, with respective rates of subsidies 54 and 28 percent. We neglect the 1% standard rate of VAT applied in 1388 because we have no information about how it was interacting with subsidies. In our post-reform period, which is year 1393 of the Iranian calendar (from March 21, 2014), we have considered that the tax base is shaped by 7% rate of VAT.

The elasticities and the fictitious demand of a class i household adjusting labor effort to earn the before-tax income of class j households that appear in ( 6) and ( 7) obtain by estimating a complete AIDS for each class. The price indexes of the COICOP categories come from the Central Bank of Iran for every province of Iran and month from 1388 to 1393. Some details about the AIDS specification 3 and the relevant elasticities are given in Appendix B.

The name of the reform stems from the Targeted Subsidies, which are fixed income transfers paid to every person to compensate for the loss in purchasing power. At the completion of the reform, these transfers coexist with a progressive income tax ruled in Iran by the Direct Taxation Act. Our data provides us neither with household before-tax nor after-tax income. We treat the before-tax income as an unobserved component of household taste heterogeneity that is perfectly correlated with the income class by estimating the AIDS for every income class separately. This gives us demand functions conditional to the before-tax income. Final consumption expenditure is used as a proxy for the after-tax income.

Empirical results

Pattern of incentives

In order to recover (λ * ij ) from ( 7), we need the fictitious consumption ξ k ij = ξ k i (q, y j , R j ). However the data contains no information about household before-tax income. From now on, we approximate ξ k ij by the fitted AIDS of type i evaluated at the after-tax income of class j, i.e., ξ k ij is set to ξ k i (q, y i , R j ). This approximation is exact when utility is separable in before-tax income, e.g.,

u i (x, y) = U i (x) -V i (y). ( 8
)
Since the AIDS is log-linear in income, imputing the mean after-tax income of class j would bias the fitted demand upwards and thus dampen the estimated magnitude of the Lagrange multipliers in [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF]. To tackle this issue we have used a matching procedure to 3 Since the AIDS budget share is log-linear in after-tax income (total expenditures), we have removed the bottom and top 1 percent of the income distribution to improve the quality of demand estimation (see footnote 1). We use family size as demographic control. The COICOP categories Alcoholic beverages, tobacco and narcotics (02), Housing, water, electricity, gas and other fuels (04), Recreation and culture (09) and Education [START_REF] Gahvari | Fiscal reforms in general equilibrium: theory and an application to the subsidy debate in Iran, The B[END_REF] are considered as fixed expenditures and treated as individual controls.

impute to each household of class i, if mimicking some type j household, the after-tax income of the household of class j who has the nearest score. 4 Appendix C reports the resulting consumption ratio differences

n i q k ξ k ij q k ξ k - n i n j n j q k ξ k jj q k ξ k (9)
that appear in [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF]. The most significant differences (in bold in Appendix C) are concentrated on Clothing, Transport and Communication. They mostly involve the poor in 1388, while all the income classes are concerned in 1393. This might reflect a widening of between-class differences in consumption tastes over the sample time window. The non negative per household multipliers (λ * ij /n ij ) solutions to [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF] are those minimizing the sum of the K squared left-hand sides of (7) in 1388 and 1393. Over the 6 incentive constraints, few actually bind. In 1388 only the incentive constraint involving the rich mimicking the poor is binding. If the government could design a specific tax of 1 rial applying to one rich household when mimicking its matched poor, the social gain would amount to 1.29 rial. We have computed (upper one-sided) confidence intervals using the non-Studentized pivotal bootstrap method. The multiplier associated with this binding incentive constraint should not exceed 5.44 rials, while all the remaining multipliers can be set to 0. This exercise allows us to identify precisely some limits to redistribution in Iran before the implementation of the Targeted Subsidies reform: the social desire to preserve the existence of the upper income class prevents the government of Iran to implement more generous income transfers toward the poor. This finding suggests political economy justifications to Ahmadinejad's Targeted subsidies program as a change in both consumption and income taxes suitably designed to favor the poor and/or hurt the rich.

In 1393 the Lagrange multipliers are very low in magnitude. The highest one is associated with the incentive constraint where the poor are ready to mimic middle income class households, with a 1 rial tax bearing on a poor yielding only 0.04 rial to the society.

We find no compelling evidence for the familiar local downward incentive pattern in our setup where agents differ according to consumption tastes and labor skill. In 1388 a non neighboring downward incentive constraint binds (the rich envy the poor but they do not envy the middle class). In 1393 the prevalent constraint is a neighboring upward constraint. To assess for the impact of the a priori restriction that only local downward incentive constraints can bind, we have reproduced the same minimization exercise under the additional requirement that the Lagrange multipliers associated with either non local or local upward incentive constraints are 0 in [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF]. The results reproduced in columns 3 and 4 of Table 1 show that the rich would then be mistakenly identified as mimicking the middle class in 1388. The magnitude of 0.7 rial of the multiplier associated with this incentive constraint is much lower than 1.29 rial: the standard pattern of local downward incentive constraints biases downwards the quantitative importance of incentives.

Redistributive impact of the reform

One can now get the profile (a * i ) by reintroducing the multipliers (λ * ij ) reported in Table 1 into [START_REF] Esfahani | Hidden public expenditures and the economy in Iran[END_REF]. The results are given in columns 1 and 2 of Table 2. The marginal utility of income α ij of class i mimicking class j = i households is approximated by the fitted after-tax income derivative of the AIDS indirect utility for a class i household receiving the after-tax income R j of its matched class j household,

α ij = ∂v i ∂R (q, y j , R j ) ∂v i ∂R (q, y i , R j ).
The approximation is exact for the class of separable preferences [START_REF] Finkelstein | Multiple dimensions of private information: evidence from the long-term care insurance market[END_REF]. The resulting ratios of marginal utilities of income α ii /α ij in (6) are given in Appendix C. Following the Iran-Iraq war, the percentage of Iranians below the poverty line started to declined from 30% to below 10% in our pre-reform period (Salehi-Isfahani, 2017). However there is little agreement about the role played by the oil-financed consumption subsidies in this decline. Indeed much of the decline occurs over a period of possibly confounding economic growth. Moreover pro-poor bread subsidies coexist with energy subsidies that tend to benefit richer households that have more cars and energy allowances (Guillaume and al., 2011; Salehi-Isfahani, 2016), which blurs the overall redistributive impact of the subsidies. The results reproduced in column 1 of Table 2 highlight a clear progressive design of taxes in line with the view of Mahmoud Ahmadinejad as a populist leader favoring the poor. We find that a 1 rial transfer to a poor household yields more than 2 rials to the society while the recovered social valuations of richer households do not exceed 1.4 rial. Note that the estimates of the social valuations in column 1 lack of precision, which possibly reflects high homogeneity of consumption tastes within income classes. The economic situation of Iran is very different in 1393, at the outcome of the reform. The Rohani government then faces economic recession associated with dampened income inequality. 5 According to Salehi-Isfahani (2017), 'for the time being, delivering on the promise of greater equity seems to be low on the agenda of policy makers in Iran [• • • ] and not much should be expected [• • • ] other than what market forces make possible.' This analysis is closely aligned with the social valuations reported in column 2. All the valuations now are very precisely estimated around 1 rial: the social welfare function of the government of Iran displays the Benthamite (unweighted Utilitarian) shape valuing identically the various types of households.

Columns 3 and 4 give the valuations obtained in the counterfactual experiment where only the local downward incentive pattern can bind. The theoretical rule (6) for optimal income taxes shows that the valuation of a mimicked type i (λ * ji > 0 for some j = i) is high compared to the situation where this type would be not envied (λ * ji = 0 for all j = i). Symmetrically the valuation of a mimicker type i (λ * ij > 0 for some j = i) is low compared to the situation where this type would envy nobody (λ * ij = 0 for all j = i). In 1388 the rich mimic the middle class in the counterfactual experiment while they actually mimic the poor. The valuation of the poor (middle income class) is accordingly adjusted downwards (upwards) from column 1 to 3. Thus the social importance of the poor is underestimated when restricting attention to the local downward incentive pattern.

Concluding comments

This paper proposes a simple new method to assess empirically the pattern of relevant incentive constraints. It relies on the estimation of the Lagrange multipliers associated with incentive constraints that appear in the first-order conditions for optimal taxes. By the usual complementary (Kuhn-Tucker) slackness conditions, an incentive constraint is binding if the associated multiplier is positive. An illustration on data from Iran over the period of the Targeted subsidies reform highlights the high social importance of the poor before the reform: a 1 rial income transfer toward a poor household yields at least 2 rials to the society. The redistribution toward the poor is however limited by the social concern of preserving a rich small elite: the incentive constraint where the rich are ready to mimic the poor appears binding before the reform. Pressures due to incentive constraints are about absent at the outcome of the reform. Greater equity then seems to be far from the concerns of the government of Iran. These findings appear consistent with political economy considerations stressed by, e.g., Salehi-Isfahani (2017).

1. Our database does not provide us with information about before-tax income. Under a suitable separability assumption on individual preferences, one can dispense from this information in the treatment of mimicking behavior. Separability sounds more plausible at the high level of aggregation of 2-digit COICOP consumption categories that we use: changes in labor supply may then mostly trigger within category substitutions, e.g., buying within the broad Clothing category some tailored-suit rather than casual wear when an individual is working. Our methodology however does not rely on separability assumptions. It would be worth to apply it on a richer database that contains before-tax income information.

2. There are two issues that would be worth to explore further in the specific case of Iran. First, anecdotal evidence suggests a weak enforcement of income taxes in the private sector. Our results should not change too much if the government of Iran designs taxes assessing accurately the induced tax leakage. Otherwise the amended first-order conditions for optimal taxes affect the recovered incentives and social valuations. The second issue concerns the production side of the economy. Our setup supposes that this side is not affected by the reform. The part of the Reform Act stipulating that some revenues raised in the reform would take the form of direct transfers to firms was never implemented. Still energy subsidies benefited firms, which should influence producer prices.

A Taxes and expenditures

Table 3 gives the expenditures for the 8 COICOP categories that are considered as flexible, the total expenditures on these categories and the corresponding budget shares.

B AIDS estimation

The elasticities and the marginal utility of income are derived from an AIDS specification estimated for different income classes, which allows for taste heterogeneity perfectly correlated with income. The AIDS specification assumes that the budget share of good k for a household h in income class i (which is her type of the theoretical setup) obeys

q k ξ k hh R h = δ k i + ζ i f h + k γ kk i log q k + β k i log R h Q i ,
where Q i is a income class i consumer price index defined by

log Q i = δ 0 i + k δ k i log q k + 1 2 k,k γ kk i log q k log q k .
Household h living in a given province and surveyed during a given month faces consumer prices indexes (q k ) computed by the Central Bank of Iran for this province and this month. The demand functions are estimated using the number of consumption units and the household demand for COICOP categories 02, 04, 09 and 10 as household h demographic control included into the 5-dimensional vector f h , and imposing the usual AIDS restrictions on parameters k >0

δ k i = 1, k γ kk i = k γ kk i = 0, k β k i = 0, γ kk i = γ k k i .
These restrictions ensure that budget shares sum to 1, and that homogeneity and symmetry properties of demand are satisfied. The demand functions are estimated for each income class using consumption data for every year from 1388 to 1393. The estimated demand functions yield the elasticities that appear in ( 6) and [START_REF] Finkelstein | Adverse selection in insurance markets: policyholder evidence from the U.K. annuity market[END_REF]. These elasticities are computed separately for year 1388 and for year 1393. They are given in Table 4.

Table 1 :

 1 Incentive pattern in Iran

			Actual	Restricted
		Tax system of year	1388	1393	1388	1393
	Actual class Imputed income
	Poor	Middle	4.68e-03	3.88e-02	0	0
			(1.71e-02)	(1.34e-01)
		Rich	1.50e-08	1.04e-06	0	0
			(5.42e-08)	(3.58e-06)
	Middle	Poor	1.17e-04	3.31e-07	4.82e-09	3.24e-02
			(4.92e-04)	(1.41e-06)	(7.95e-09)	(3.29e-02)
		Rich	2.77e-10	1.10e-08	0	0
			(1.13e-11)	(4.65e-08)
	Rich	Poor	1.29	8.17e-10	0	0
			(5.44)	(3.37e-09)
		Middle	1.06e-06	4.01e-06	7.06e-01	1.38e-09
			(4.47e-06)	(1.82e-05)	(7.46e-01)	(2.41e-09)
	1. Poor (D1-D3); Middle (D3-D9); Rich (D9-D10).
	2. Upper one-sided 1000-iteration bootstrap confidence interval at the 90 percent level.

Table 2 :

 2 Social valuations in Iran (a * i )

	Tax system	Actual	Restricted
	Year	1388	1393	1388	1393
	Income class i 1
	Poor	2.37	0.91	1.36	1.01
		(1.96 -3.7) 2	(0.84 -0.99)	(1.36 -1.36)	(1.01 -1.01)
	Middle	1.29	1.04	2.03	0.97
		(0.30 -1.4)	(0.98 -1.21)	(2.00 -2.08)	(0.97 -0.98)
	Rich	0.95	0.98	0.95	0.98
		(0.41 -1.43)	(0.88 -0.98)	(0.94 -0.97)	(0.98 -0.98)
	1: Poor (D1-D3); Middle (D3-D9); Rich (D9-D10).
	2: Two-sided 1000-iteration bootstrap confidence interval at the 90 percent level.

Table 3 :

 3 Taxes and expenditures1

		Food	Clothing Furnishings Health Transport Communication Restaurants Other	Total 2
	Taxes
	Year 1388 3	-0.28	-0.14	-0.23	-0.26	-0.54	0	0	-0.09
	Year 1393	0	0.07	0.07	0	0.04	0	0.07	0.07
	Expenditures
	Poor 4	66,699	5,100	4,105	5,897	9,645	7,172	1,143	4,630	104,394
		(63.9) 5	(4.8)	(3.9)	(5.6)	(9.2)	(6.9)	(1.1)	(4.4)
	Middle 4	79,245	12,036	5,465	9,705	12,886	8,461	2,203	5,905	135,909
		(58.3)	(8.9)	(4)	(7.1)	(9.5)	(6.2)	(1.6)	(4.3)
	Rich 4	112,052	31,392	9,653	23,669	20,079	11,802	4,660	9,028	222,339
		(50.4)	(14.1)	(4.3)	(10.6)	(9)	(5.3)	(2.1)	(4)
	1. Expenditures are expressed in thousands of rials.
	2. Average household expenditures (other than COICOP categories 02, 04, 09 and 10) used as a proxy for total net income.
	3. Computed using subsidies in Gahvari and Taheripour (2011).
	4. Poor (D1-D3); Middle (D3-D9); Rich (D9-D10).
	5. Household budget share in percent.

Table 4 :

 4 Compensated price and income elasticities

	Price of	Food Clothing Furnishings Health Transport Communication Restaurants Other Income
	Rich
	1388
	Food	-0.48	0.33	0.12	0.08	-0.04	-0.08	0.07	0.01	0.74
	Clothing	0.29	-0.46	0.20	0.33	-0.18	-0.16	-0.03	0.01	1.18
	Furnishings	0.34	0.67	-1.08	0.33	-0.26	-0.12	0.07	0.05	1.05
	Health	0.11	0.49	0.15	-0.80	0.04	-0.08	0.04	0.04	1.20
	Transport	0.19	0.88	0.40	-0.15	-1.48	-0.07	0.06	0.17	1.20
	Communication	0.37	0.82	0.18	0.28	-0.07	-1.58	-0.02	0.02	1.26
	Restaurants	1.58	-0.76	0.56	0.72	-0.31	0.09	-1.35	-0.53	1.12
	Other	1.88	1.52	2.77	5.35	-6.33	-0.91	-3.80	-0.49	-1.91
	1393
	Food	-0.46	0.34	0.12	0.08	-0.05	-0.11	0.06	0.02	0.77
	Clothing	0.35	-0.47	0.21	0.30	-0.17	-0.19	-0.04	0.01	1.18
	Furnishings	0.39	0.65	-1.06	0.30	-0.25	-0.14	0.06	0.05	1.05
	Health	0.14	0.49	0.16	-0.84	0.06	-0.11	0.04	0.05	1.23
	Transport	0.24	0.90	0.41	-0.20	-1.50	-0.10	0.06	0.19	1.21
	Communication	0.42	0.70	0.17	0.24	-0.07	-1.47	-0.01	0.03	1.19
	Restaurants	2.27	-1.37	0.78	0.92	-0.42	0.16	-1.55	-0.80	1.18
	Other	0.77	0.72	0.83	1.61	-1.74	-0.34	-1.00	-0.86	0.23
	Middle
	1388
	Food	-0.41	0.34	0.10	0.00	0.01	-0.10	0.06	0.00	0.81
	Clothing	0.30	-0.42	0.14	0.15	-0.11	-0.12	-0.01	0.07	1.14
	Furnishings	0.44	0.73	-1.06	0.12	-0.24	-0.07	0.03	0.06	0.97
	Health	0.00	0.56	0.09	-1.07	0.28	0.01	0.03	0.11	1.10
	Transport	-0.11	1.11	0.47	-0.74	-1.29	0.14	0.07	0.36	0.79
	Communication	0.58	0.80	0.09	-0.02	0.09	-1.54	-0.04	0.04	1.03
	Restaurants	3.00	-0.52	0.28	0.40	-0.37	0.33	-1.75	-1.37	1.27
	Other	0.07	1.72	0.27	0.70	-0.87	-0.15	-0.63	-1.11	0.73
	1393
	Food	-0.40	0.36	0.10	0.00	0.01	-0.13	0.05	0.01	0.83
	Clothing	0.35	-0.42	0.15	0.14	-0.11	-0.16	-0.02	0.08	1.14
	Furnishings	0.48	0.72	-1.05	0.11	-0.23	-0.11	0.02	0.06	0.97
	Health	-0.01	0.58	0.09	-1.11	0.31	-0.01	0.02	0.12	1.11
	Transport	-0.08	1.13	0.47	-0.76	-1.29	0.11	0.07	0.37	0.78
	Communication	0.56	0.70	0.09	0.01	0.05	-1.43	-0.03	0.04	1.02
	Restaurants	7.78	-2.33	0.63	0.88	-0.96	1.03	-3.14	-3.88	1.75
	Other	0.18	1.45	0.23	0.56	-0.68	-0.16	-0.49	-1.08	0.79
	Poor
	1388
	Food	-0.34	0.29	0.10	0.07	-0.02	-0.18	0.06	0.01	0.91
	Clothing	0.40	-0.56	0.12	0.18	-0.09	-0.19	0.02	0.12	1.12
	Furnishings	0.57	0.50	-0.88	0.23	-0.18	-0.14	-0.11	0.01	0.98
	Health	0.24	0.45	0.14	-1.01	0.21	-0.10	-0.01	0.09	1.05
	Transport	0.18	0.72	0.35	-0.66	-1.00	0.01	-0.05	0.45	0.90
	Communication	0.62	0.49	0.09	0.10	0.00	-1.37	0.04	0.04	1.06
	Restaurants	1.19	0.34	-0.38	-0.05	0.09	-0.22	-0.58	-0.39	1.13
	Other	0.20	1.30	0.02	0.37	-0.60	-0.15	-0.30	-0.84	0.92
	1393
	Food	-0.32	0.30	0.10	0.07	-0.02	-0.21	0.06	0.02	0.91
	Clothing	0.43	-0.56	0.12	0.17	-0.09	-0.22	0.02	0.13	1.12
	Furnishings	0.60	0.50	-0.88	0.22	-0.18	-0.17	-0.11	0.01	0.98
	Health	0.26	0.45	0.15	-1.03	0.22	-0.12	-0.01	0.10	1.05
	Transport	0.20	0.72	0.36	-0.67	-1.00	-0.02	-0.05	0.46	0.90
	Communication	0.64	0.47	0.09	0.10	-0.01	-1.36	0.04	0.04	1.05
	Restaurants	1.27	0.34	-0.41	-0.08	0.10	-0.25	-0.55	-0.42	1.14
	Other	0.27	1.18	0.03	0.33	-0.53	-0.18	-0.26	-0.85	0.93

We remove observations in the bottom and top 1 percent of the income distribution, and those where more than 99 percent of the household budget is allocated to one consumption category only. Our final sample consists of 77,364 observations.

Details about the COICOP classification are available at http://unstats.un.org/unsd/cr/.

The score is computed for each year 1388 and 1393 separately from a logit model that regresses the probability of belonging to class i rather than j on age, education, household size and province.

International sanctions hurt top incomes while Targeted Subsidies transfers represent a larger share of the income of the poor. In our (urban) sample the Gini coefficient of total expenditures decreased from 0.33 in 1388 to 0.28 in 1393.

C Mimickers' behavior

The demand of a household of class i that would mimic the behavior of a household of class j is computed from the fitted demand of class i evaluated at the after-tax income (total expenditures) of the nearest score household of class j. The differences [START_REF] Finkelstein | Testing for asymmetric information using 'unused observables' in insurance markets: evidence from the U.K. annuity market[END_REF] are given in Table 5. The last column of this table gives the corresponding differences in marginal utilities of income in the ratio formulation relevant in [START_REF] Esfahani | Hidden public expenditures and the economy in Iran[END_REF].

Table 5: Mimickers' fictitious demand


Alexander Sigman

Matthias Megyeri

Nicolas Misdariis

alarm/will/sound: A Multidisciplinary Research/Installation Project

is a research--based interdisciplinary project, currently being developed by Korea

Audio design and auto design-a mutual regard

The convergence of sound engineering and auto design is nothing new. Whether refining the acoustics of the sedan cabin, installing the optimal speaker--system, enhancing the invigorating sound emitted from the engine upon ignition, or selecting the tone assigned to a remote "en-try system," the auditory aspect of the driving experience is treated with great sensitivity. Whenever a driver or (especially) a mechanic detects an internal malfunction, a degree of sophistication is required to isolate a given sound from a far more complex "scene," a veritable "or-chestra" of sound--structures.

However, there are two sound--emitting devices for which little variation from one vehicle to the next is evi-dent: horns and security systems. While there have been recent advances in car horn design (Lemaitre et al 2009), a projected signal is meant in all cases to be unambigu--ous...and unambiguously directed towards a particular party. But the matter becomes more intricate when ad-dressing alarms. To whom is the alarm --signal directed? The potential perpetrator? The owner? Passers-by/police? How is one to respond exactly? One may iden-tify two common reactions: 1) such alarms are irritating, and continue for too long; 2) and are therefore often (to be) ignored.

There does exist a general tendency in designing any security system to prohibit the pure functionality of the system from being compromised or dwarfed by aesthetic considerations. Yet, the "repertoire" that typifies the standard car--alarm has established an iconic presence in itself in popular culture, divorced from a particular func-tion. A brief survey of Youtube videos, for instance, will reveal countless examples of footage of people dancing to alarms. Although one may dismiss such actions as mildly comedic novelty acts, they do reveal a more seri-ous implication, often taken for granted: parked in a pub-lic setting, the "private space" consists of not only the interior of the vehicle...but also extends to a rather wide radius around the car-i.e., even standing in close prox-imity to the vehicle, especially when an alarm has been activated, may be perceived as suspect.

In undertaking the current project, we have been explor-ing this delineated boundary between public and private space, as well as the influence of alarm--systems upon human behavior to a greater extent. What if the passerby's position in relation to the car-or a change in any other physical parameter-could alter one (or more) of the alarm's sound parameters? Would a security sys-tem be capable of inviting/engaging the unsuspecting pedestrian in an intimate, car--internal listening experi-ence, rather than deterring such an intrusion? If an alarm--repertoire is (seemingly) incessant, but sufficiently interesting, will one modulate one's mode of listening, substituting a set of expectations proper to a concert-situation, rather than instinctively ignoring the alarm or avoiding the vehicle's location?

Besides the exploration of car--alarm/human interac-tions, questions regarding the identity of the alarm sys-tem, as an object, a product, in and of itself, have arisen. If the basic apparatus of the product is maintained, but its function has been modified or radically shifted, does its integrity as a product abide? Or does this mere "iso-morphic" relationship to the alarm model constitute a complete redefinition of the object? That it so say: wherein lies the alarm system's identity-its hard-ware/components, the sounds projected, or the re-sponses elicited from those within the vehicle's vicinity?

Artist Backgrounds and Motivations

As a composer, Alexander Sigman has recently been in-terested in the influence of the sounds of physical envi-ronments on the aesthetics of composers and sound art-ists, as well as the impact of composers and sound artists on physical environments. Many of his recent ensemble, electroacoustic, installation, and media works deal with the reconstruction of, interaction with, and importation of sonic source materials from urban and industrial envi-ronments. Over the past several years, he has collabo-rated with filmmakers, video artists, visual artists, sculp-tors, and dancers on a diverse array of projects.

In addition to his compositional activities, Sigman also has a background in Cognitive Science-Music Cognition and Timbre Perception in particular-and has thus been approaching the project from both an artistic and re-search perspective.

Coming from a political poster design and public art background, Stuttgart/London--based artist Matthias Megyeri is interested in how the human mind produces practical solutions for the problems of security as well as how one can create an alternative model by still keeping the conflict between power and control alive in the de-sign process. This question may seem to be limited to the economy of industrial design, but through his multi-layered research and conceptual artistic approach the cultural, social, and psychological aspects of protection, and privacy comes out in accordance with how the lan-guage of security is visually articulated.

As a Stuttgart native, Megyeri has also maintained a long--standing interest in directly engaging with the au-tomobile industry dominant in the region (Stuttgart is home to both Mercedes and Porsche)-with respect to both car design practices and relationships between car manufacturers and artistic institutions and artists.

IRCAM and the Auto Industry: Research Precedent

In recent years, the IRCAM Sound Perception and Design (SPD) team has been engaged in several research pro-jects pertaining to both environmental and automobile sound design and perception. In 2009, a study on car horn sound quality was conducted (see reference above) that aimed at providing perceptually--based specifica-tions for creating new sounds for car horn. In the same period, another research project dealt with a methodo-logical question: the perceptual influence of instruction modality (verbal vs. visual) on identification and judg-ments of car interior sonic events (Susini, Houlx, Misdar-iis et al. 2009). In the Human--Computer Interface (HCI) field, the team has also contributed several significant publications, including studies on: a) the influence of audio features on the perceived urgency and its design

Car Alarm Origins

The first car alarm patent dates from 1919. The inven-tion consisted of a switchboard for enabling and disa-bling the alarm (either the existing car horn, or an addi-tional electric horn), and a mechanism for opening the ignition circuit for preventing car operation. The control-ling switchboard functioned like a combination lock: the driver threw a certain number of switches, with no visi-ble change to the outward appearance to the switch-board. If a perpetrator were to reset the incorrect switches (assuming that the device is not concealed), or to crank the car or start the battery without disabling the alarm, the horn would sound.

An alarm invented nearly a century ago, prior to the introduction of the key--operated ignition circuit, may seem primitive at first blush. Nonetheless, the combina-tion lock model upon which the alarm--activation inter-face was based provides a compelling model for possible car alarm interaction design innovations-whether tak-en literally or loosely interpreted (see Section 10 of this paper).

Alarmingly Useless: Problems with the "Car

Alarm as Deterrent" Model

Why introduce innovations to a purely functional device that is ostensibly straightforward in its design? In a 2003 report analyzing the benefits, costs, and effectiveness of car alarms in New York City (Friedman et 'The thief knew that nobody in the neighborhood would pay the slightest attention to a car alarm,' he says, 'so he used the noise to cover the sound of breaking the window. Then he stole the radio out of the car.'" 5

Extant Alternatives to the Audible Alarm

Besides devices such as brake and wheel locks and pas-sive immobilizers, the Lojack Stolen Vehicle Recovery System has proven to be a highly successful alternative to the standard audible car alarm. Rather than emitting tones, the Lojack system is a radio transceiver installed in various locations in vehicles that employs GPS tech-nology to track a vehicle in the event of a reported theft. As many stolen cars are destined for the "headquarters" of professional auto theft crime rings, this device ena-bles authorities to prevent future car theft and locate additional (unreported) stolen vehicles.

While the objective of our project is to modify and ex--tend audible alarm systems, it would be of interest to utilize embedded GPS in designing interaction models (see Section 10).

Other (relatively) successful deterrence strategies that have informed our research derive from the public space security domain. In the past decade, there have been campaigns to curb anti--social behavior amongst younger people congregating in urban centers-particularly in the UK and Netherlands. The so--called Mosquito anti-loitering device is one such innovation emerging from such campaigns (see Figure 2). The Mosquito emits a high frequency (17 KHz.) that lies outside of the audible range for most adults above the age of ca. 25. For those within the age range of 13--25, the painful sound level at which the frequency is emitted (theoretically) causes crowds of loitering teenagers to disperse.

Another targeted dispersal method involves the projec-tion of classical music recordings into public spaces.

Comforting to all but individuals exhibiting anti--social behaviors, there has been evidence of such individuals avoiding these designated areas. Needless to say, this practice brings up a host of sociological concerns regard-ing the relationship between security and music aesthet-ics, the use of music as a defensive weapon, the perpet-uation of associations of classical music with social class and authority, and public/private space boundaries.

One aspect of our investigation of social behaviors vis--à-vis alarm systems will entail an inversion of this music-as--deterrent protocol. As is elaborated in Section 9, one category of our alarm prototype sound corpus consists of synthetic fragments typical of digital electroacoustic music, but strange or unfamiliar to most listeners. We intend to analyze responses to this sonic material when it is presented in the context of a car alarm system's repertoire. Will it be possible through the expansion of the alarm's sonic vocabulary in this manner to encour-age close listening and interaction with the alarms?

Car Alarm Communication Model

From our perspective, the alarm system may be regard-ed not only as a functional deterrent, but also as a nodal point of interaction. In the context of alarm/will/sound, the alarm prototypes to be developed have been con-ceived as both instruments, activated and manipulated by humans, and as intelligent "interlocutors," sensing and responding to changes in the environment, and adapting their behavior patterns accordingly. As such, the alarm's relationship to society expands from a fixed one--way communication stream to a fluid, environment-specific dialogue. The alarm prototypes will thus bear the potential to expand the usual repertoire of modes of deterrence, while introducing several modes of user engagement.

Project Phases and Tasks

The project has been divided into two primary phases: 1) conception, design, and production of sound libraries and interactive systems; 2) refinement and validation of results.

Phase I (January--August 2013):

The first objective in this phase of the project was to define the functionality of the alarm systems. In keeping with our "interactive instrument" alarm model, two sali-ent alarm functions were established: 1) perpetrator deterrence and user engagement via an expansive, un-predictable, and time--varying catalogue of individual sounds and sound--complexes; 2) detection and commu--nication of local (physical and cultural) environmental information to the user.

With these functions in mind, a sound library taxonomy was constructed (see Figure 3). Representative sounds for each category and sub--category in the taxonomy were selected, generated, edited, and tagged with se-mantic descriptors. In parallel, interaction and hardware designs were drafted, and necessary materials and pos-sible hardware configurations determined (see below).

Phase II (September 2013--February 2014):

The experimental component of the project has re-volved around the sound library produced in the first part of the project. Experiments are being designed and conducted based upon both the composer's (A. Sigman) and the product designer's (M. Megyeri) intuition and expertise, together with preliminary formal knowledge on alarm sound morphology/functionality and everyday sound perceptual structures. The main goal of the exper-iments is to define semantic and/or acoustic categories relevant to organizing the original sound library, as well as to estimate physical or perceptual distances between actual alarm sounds and the non--standard sounds in-cluded in the corpus. Further expansion and refinement of this corpus will subsequently be informed by the re-sults of these experiments.

Two main sets of experiments are being carried out. The first tests for semantic category membership. The se-cond pertains to alarm type designation and urgency ratings. The results of both will inform the generation of semantic and acoustic descriptors with which the corpus will be tagged. The experimental protocols and goals are described in greater detail below.

Sound category membership experiments. The ultimate objective of this first series of experiments is to generate a perceptual characterization of the sound corpus librar-ies with regards to criteria such as sound source identifi-cation (identifiability), causal (un)certainty, and sound typicality [START_REF] Ballas | Common Factors in the Identification of an Assortment of Brief Everyday Sounds[END_REF]. Mainly for scientific and experi-mental reasons, these experiments are being carried out using a restricted set of sounds derived from the corpus and will be deployed by means of an online protocol in order to collect a large amount of results. 6

Subsequently, other criteria, such as sound--source origin, time--period, context, mode of production, mate-rials, type of impact, temporal morphology, rate and density, underlying movement, physical state, and dura-tion could be defined and/or experimentally studied in order to tag as precisely as possible the original corpus.

In parallel, the computation of acoustic features on the studied sounds is being conducted in service of con-structing a computational model that will be able to pre--cisely predict sound category membership of a new item when introduced to the corpus.

Alarm type designation and urgency ratings. The level of perceived urgency of a set of extended composite alarms assembled from the different sounds in the cor-pus together with perceptual distances between these new alarm sequences and prototypical alarm repertoires will be experimentally measured in order to obtain new information on the corpus and be able to associate its components with the basic modes of deterrence and user engagement (interaction) defined within the scope of the project.

Sound Categories, Parameters, and Sources Categories and Sound Parameters

As is indicated in Figure 3, three primary sound catego-ries were devised: 1) real alarm sounds; 2) individual sounds; and 3) "auditory scenes," or sound complexes of various durations composed of multiple individual sounds. Category (3) was then subdivided into a) record-ed and b) synthetic sounds.

Individual sounds were grouped according to a) context; b) sound source semantic category; c) sound source physical material; d) sound source physical mode of production and e) sound source acoustic property de-scription. Acoustic properties include attack--type, tem-poral envelope, spectral envelope, and (relative) dura-tion.

As such, the Individual Sound taxonomy becomes rather complex. At the highest level in this taxonomy, constitu-ent sounds were segmented into Synthet-ic/Electroacoustic, Natural/Vocal, Mechani-cal/Industrial, and Film Danger Icon categories.

The Synthetic/Electroacoustic category consists of sounds created via synthesis techniques typical of elec-troacoustic music of the past 40--50 years, as well as audi-tory illusions (e.g., Shepard glissandos and monaural beats). Human and animal vocalizations-primarily of a threatening, urgent, or otherwise attention--attracting nature-are included under the Natural/Vocal rubric. As the most multifaceted category, the Mechani-cal/Industrial rubric is constituted by electrical and me-chanical tool, appliance, and vehicle sounds (trains, boats, airplanes, and helicopters), as well as explosions, gunshots, impacts, and ricochets. Film Danger Icons are clichéd (or at least highly recognizable) tension/danger cues found in classic and modern cinema scores. Finally, the Real Alarm Sounds category consists of six entries belonging to a standard car alarm cycle-in sequence, and in combination with an iterated car horn.

In all cases, sounds that were too similar to auto/traffic sounds, were likely to be masked by the vehicle's physi-cal environment, or did not draw adequate attention to themselves (e.g., ambient natural sounds) were avoided.

Sound Sources

Among the non--synthetic sounds in all three primary categories, the majority were mined from existing sound databases (e.g., SoundIdeas, Blue Box, Auditory Lab [http://www.psy.cmu.edu/~auditorylab/website/index/h ome.html], and freesound.org). Under the "Auditory Scenes--Recorded Sound" rubric, a series of field record-ings of public spaces in Paris-streets, the Forum Les Halles shopping concourse, the Centre Pompidou, metro stations, and train car interiors-were compiled in Feb-ruary 2013. It is intended that the collection of field re-cordings be expanded over time to include further site-specific entries, both by the authors and by alarm proto-type users (see below). Synthetic individual sounds were generated and edited using such synthesis software as AudioSculpt, Pure Data (Pd), SuperCollider, and the Python--based concatenative synthesis program Audioguide (authored by composer Ben Hackbarth).

Interaction Design Models

There will be three basic modes of interaction associated with our alarm prototypes: enabling/disabling, triggering, and data transmitting. Extending the "combination lock" functionality of the earliest patented car alarm (dis--cussed in Section 4), we have been experimenting with physical enabling/disabling codes: a sequence of gestures on the surface of or in close proximity to the vehicle de-tected by a sensor network.

Activation--type will vary according to three possible alarm triggering settings: 1) self--triggered; 2) time-constant, passerby--triggered; and 3) time--varying, passerby--triggered. In the self--triggering scenario, the alarms will be set off at irregular intervals, independent of the actions of the passerby. For the time--varying set-ting, changes in sound quality, type (i.e., sound library entry), or spatial location will correspond to changes to any detectable physical parameter associated with the passerby (e.g., proximity, position with respect to the vehicle, orientation, angle, and speed).

As a supplement to these standard alarm functions, we will repurpose the GPS technology at the core of Lojack and other systems in order to sonically convey localized environmental data to the vehicle driver. Such data will include (but not be limited to) weather conditions, urban "auditory scenes," and location--specific musical samples and acoustic information.

Expected Project Outcomes

Given the authors' activity in and interplay with artistic, research, and industrial domains, it is intended that the research and development of car alarm prototypes a) be presented in site--specific installation contexts; b) inform future research pertinent to security systems and car sonification; and c) apply to car industry innovations. The installations will be designed according to two basic scenarios: a) as a series of public space "interventions" and b) as an interactive exhibition in a gallery space. In the former situation, alarm prototypes will be embedded in vehicles parked throughout an urban environment. These alarms will be alternately triggered by passersby and self--triggered, and responsive to physical and envi-ronmental changes. In the latter, the prototypes will pos-sess the same functionality as in the public space scenar-io, but also enable exhibition visitors to control, manipu-late, and expand the sonic repertoire and patterns emit-ted by the alarms. Visitors will be able to search the sound database via entering descriptive terms at a cen-tral console and select individual sounds or construct "auditory scenes." In addition, they will be requested (via an online platform) to upload recordings derived from their respective physical environments for possible inclu-sion in the sound corpus. Over the course of the exhibi-tion, visitors' experiences with the various alarm interac-tion models will be documented via video recording, data collection at the central console, and questionnaire re-sponses.

In recent years, certain high--end automobile manufac-turers (e.g., Porsche and Rolls--Royce) have customized features of their models according to customer specifica-tions. Given the experimental and idiosyncratic nature of our project, it is expected that our alarm prototypes will bear more potential "bespoke" than mass--market appli-cations. However, given the ever--increasing integration of mobile technologies, the need for creative solutions for (mass market) car companies to differentiate them-selves from their competitors, it is not unimaginable that aspects of these prototypes-be it the interactive system designs or sound corpus entries-could be adapted by auto industry sound design teams.

Conclusion

By February 2014, it will be highly unlikely that the alarm/will/sound project will have attained completion. Rather, it is our intent to lay a firm foundation-through design, production, and experimentation-for continued expansion and refinement. The constraints and specifications applied to hardware design of the alarm prototypes-a dimension of the project not thoroughly addressed in this paper-will largely be determined by the nature of our collaboration with the auto industry.

Figure 1 .

 1 Figure 1. From top to bottom: Matthias Megyeri's Sweet Dreams Secu-rity ® bear--shaped padlock; Megyeri's animal picket fence; examples of security devices at the boundary of public and private space; examples of cute objects at the boundary of public and private space.

  application to several car interior Human--Machine Inter-faces (HMI) (Suied et al. 2008); and b) different sonifica-tion approaches of an automotive media--center hierar-chical menu and its evaluation in terms of navigation performance (Misdariis et al. 2011). More recently, IRCAM's partnership with the French automobile manu-facturer Renault, in conjunction with composer Andrea Cera, on sound design for electric vehicles (one com-mercialized model and several concept--cars) (Misdariis et al. 2012), has received significant publicity. This pro-ject has led to complementary works--in--progress in this domain such as a recent study of electric vehicle detect-ability in urban environments (Misdariis et al. 2013). alarm/will/sound follows in this line of timbre-perception--research--driven vehicle sound design. Be-sides adapting methodologies proper to previous IRCAM researches, we are drawing upon the extensive auditory warning literature-both specific to automobile auditory signals (Yamauchi et al. 2004) and to more general cate-gories of auditory signal (Edworthy & Hards 1999; Ku-wano et al. 2007; Petocz, Keller & Stevens 2008; Stan-ton & Edworthy 1998). In addition, studies in sound clas-sification (Claude 2006; Michaud 2012) and auditory display (Barrass 1997) have informed our research and design approaches.

Figure 2 .

 2 Figure 2. The Mosquito. (http://movingsoundtech.com/our-products/mosquito--mk4--with--multi--age)

Figure 3 .

 3 Figure 3. Sound library taxonomy.