Preliminary and Incomplete. Not For Distribution. Efficient Contracting in Network Markets

Transcription

1 Preliminary and Incomplete Not For Distriution Efficient Contracting in Network Markets Darrell Duffie and Chaojun Wang Stanford University June 4, 2014 Astract We model argaining in over-the-counter network markets over the terms and prices of contracts. Of concern is whether ilateral non-cooperative argaining is sufficient to achieve efficiency in this multilateral setting. For example, will market participants assign insolvencyased seniority in a socially efficient manner, or should ankruptcy laws override contractual terms with an automatic stay? We provide conditions under which ilateral argaining over contingent contracts is efficient for a network of market participants. Examples include seniority assignment, close-out netting and collateral rights, secured det liens, and leverage-ased covenants. Given the aility to use covenants and other contingent contract terms, central market participants efficiently internalize the costs and enefits of their counterparties through the pricing of contracts. We provide counterexamples to efficiency for less contingent forms of argaining coordination. JEL Classifications: D47, D60, D70, G12, K22 Keywords: Over-the-counter, network, contracting, argaining, efficiency, game theory Duffie is at the Graduate School of Business, Stanford University, is an NBER Research Associate, and has a potential conflict of interest as a consultant to the Lehman estate. Wang is in the Doctoral Program of the Stanford Statistics Department. This version is preliminary and incomplete. We are grateful for early conversations with Martin Oehmke, David Skeel, and Jeff Zwieel. Comments are welcome.

2 1 Introduction We propose a theory of ilateral argaining over the terms and pricing of contingent contracts in a network setting. We state conditions under which contingent ilateral contracting is socially efficient, suject to the availale sets of contracts. We provide counterexamples to efficiency in settings with less effective forms of argaining coordination. In this preliminary version, we focus on a simple network of three market participants. An example application depicted in Figure 1 is the contracting etween a detor firm and a creditor, and etween the same detor firm and a derivatives counterparty. The unique tremlinghand perfect equilirium in our asic alternating-offers contingent contract argaining game specifies socially efficient actions y the three firms. Efficiency arises through the aility of the detor to internalize the costs and enefits of its two counterparties through the pricing of contracts with each of them. For example, if a particular change in contract terms would have large enefits for the swap counterparty, and could e accommodated at a small total cost to the detor and creditor firms, then this change in contract terms will e chosen in the course of pairwise contingent contract argaining, given that the detor firm can extract a compensating payment from the swap counterparty that provides a sufficient incentive to the detor and creditor. loan swap creditor detor swap counterparty Figure 1: An illustrative three-firm financial contracting network. An illustrative issue of concern in this setting is whether the equilirium contract terms would efficiently assign recovery priority to the creditor and the swap counterparty in the event of the detor s insolvency. Assignment of priority in either direction is currently permitted under U.S. ankruptcy law through exemptions for qualified financial contracts such as swaps, repos, clearing agreements, and securities lending contracts. This exemption allows for enforceale ipso facto 1

3 clauses that assign the right to terminate a contract in the event of insolvency and to liquidate collateral. This sort of clause is standard in current swap and repo contracts. There has een a deate over allowing qualified financial contracts to include ipso facto clauses, unrestricted y ankruptcy law. Roe 2013 suggests that contractual assignment to swaps of the right to terminate and keep collateral in the event of the detor s insolvency should e unenforceale or suject to rejection under an automatic stay. This issue is modeled in a different setting y Bolton and Oehmke 2014, who instead assume price-taking competitive markets and rule out the negotiation of covenants regarding the assignment of priority. Like Bolton and Oehmke 2014, our model does not cover externalities such as firesales, a tradeoff discussed y Duffie and Skeel In our model, each pair of directly connected firms argains over contractual terms. In the example setting illustrated in Figure 1, the equilirium contract prices reflect the relative distress costs of the two counterparties, allowing the detor to efficiently internalize its counterparties distress costs and assign contractual priority efficiently. For example, if the creditor suffers greater distress from loss of default priority than does the swap counterparty, then in the naturally selected equilirium, the pricing of the swap contract will include a price concession that is sufficient to convince the swap counterparty to give up priority. The creditor would in this case e willing to accept a lower interest rate in order to receive effective seniority. Conversely, if the creditor is etter equipped to suffer losses at the detor s default than the swap counterparty, then in equilirium the detor will offer a high enough interest rate to the creditor to encourage the creditor to agree to loss of priority, and a compensating higher upfront payment from the swap counterparty. The detor s shareholders have no direct concern with seniority at the detor s own default, and are therefore ale to act as a conduit y which the creditor and the swap counterparty can indirectly compensate each other for priority assignment. Our results are ased on an extension to network settings of the alternating-offers argaining game of Ruinstein Bargaining is conducted y each pair of directly connected nodes, which we call firms. Our model allows for incomplete information. While a given pair of firms is argaining, they are unaware of the argaining offers and responses eing made elsewhere in the network. In order to isolate natural equiliria, we therefore extend the notion of tremling-hand perfect equilirium of Selten 1975 to this network argaining setting. The tremling-hand perfect 2

4 equilirium choices are socially efficient y virtue of the assumed aility to sign contracts whose terms are contingent on the terms of other contracts. For instance, in the setting of our illustrative example, the creditor and the detor can choose to make the terms of their loan agreement contingent on the terms of the swap contract chosen y the detor at its swap counterparty. The efficiency of the tremling-hand perfect equilirium contract terms does not depend on some aspects of the argaining protocol, such as which pair of counterparties writes contingent contracts and which pair of counterparties uses only simple contracts. In practice, covenants in a given contract normally restrict the terms of future contracts, ut our setting is static. We also show that equilirium contract prices converge, as exogenous reakdown proailities go to zero, to those associated with the unique cooperative-game argaining solution that satisfies two proposed axioms, multilateral staility and ilateral optimality. In particular, the non-cooperatively argained prices do not depend on which pair of firms writes contingent contracts. We are working on extensions of these results to more general types of networks, under conditions that rule out general cross-network externalities. An extensive literature on network argaining games includes some prior work that, like ours, focuses on non-cooperative ilateral argaining. Until now this literature has studied settings in which there are two key impediments to socially efficient outcomes: i general cross-network externalities, and ii coordination failures that arise from a restriction to contracts that are not contingent on other ilateral contracts. We assume an asence of general network externalities and we allow contracts to have unlimited cross-contract contingencies, such as covenants. These ideal conditions are not to e expected in practical settings. We nevertheless elieve that it is valuale to characterize a theoretical enchmark network market setting in which ilateral contracting is socially efficient, suject to the restrictions imposed y the feasile sets of contractile actions. Even in our ideal setting, our analysis suggests that apparently reasonale changes to our proposed argaining protocol can lead to additional equiliria that are not efficient. Several papers provide non-cooperative ilateral argaining foundations for the Myerson-Shapley outcomes and values, as defined y Myerson 1977a. In the first of these papers, Stole and Zwieel 1996 provide a non-cooperatove foundation for the Myerson-Shapley values as those arising in the unique su-game perfect equilirium of a network game in which a firm negotiates ilateral 3

5 laor contracts with each of its potential workers. Their argaining protocol, like ours, is ased on the Ruinstein alternating offers game. In their case, however, reakdown in a given ilateral argaining encounter results in a re-start of the argaining of the firm with other workers, in which any previously agreed laor contract is discarded. In this sense, the laor contracts are noninding. The work of Stole and Zwieel 1996 is extended to more general settings y de Fontenay and Gans In a different setting, Navarro and Perea 2013 provide a ilateral argaining foundation for Myseron values, with a sequential ilateral argaining protocol in which pairs of linked nodes argain over their share of the total surplus created y the connected component of the graph in which they participate. 2 Primitives We egin with a simple three-firm network. Firms 1 and 2 argain over the terms of one contract. Firms 2 and 3 argain over the terms of another contract. The contracts specify the actions to e taken y each firm. Firm i takes an action in a given finite set S i. For each action s 2 S 2 of the central firm, there is a limited suset C 1 s 2 S 1 of feasile actions y firm 1, and a limited set C 3 s 2 S 3 of feasile actions for firm 3. That is, C i is a correspondence on S 2 into the non-empty susets of S i. Each pair of contracting firms is ale to exchange a monetary payment. Firm 2 pays firms 1 and 3 the amounts y 1 and y 3 respectively. These amounts are any real numers, positive or negative. Equivalently, firm i pays firm 2 the amount y i. In summary, firms choose actions s S = S 1 S 2 S 3 and compensation amounts y R 2, with respective quasi-linear utilities u 1 y, s = f 1 s 1, s 2 + y 1 u 2 y, s = f 2 s 1, s 2, s 3 y 1 y 3 1 u 3 y, s = f 3 s 2, s 3 + y 3, for some f 1 : S 1 S 2 R, f 2 : S R, and f 3 : S 2 S 3 R, as illustrated in Figure 2. It is important for our efficiency results that a firm s utility depends only on its direct compensation 4

6 and on the actions of itself and its direct ilateral counterparty. The condition s 3 Cs 2 and the dependence of f 3 s 2, s 3 on s 2 nevertheless imply that contracting etween firms 1 and 2 over s 1, s 2 has an influence over firm 3 through the choice of s 2. A symmetric situation applies to contracting etween firms 2 and 3. Firm 1 action: s 1 C 1 s 2 u 1 y, s = f 1 s 1, s 2 + y 1 Firm 2 action: s 2 S 2 u 2y, s = f 2s 1, s 2, s 3 y 1 y 3 Firm 3 action: s 3 C 3s 2 u 3y, s = f 3s 2, s 3 + y 3 Figure 2: Actions and utilities in the three-firm financial contracting network. In the event of a failure to reach contractual agreement, there are some pre-arranged outside options, which can e viewed as the status quo. We let s 0, y 0 S R 2 e the status-quo actions and payments. Without loss of generality via a normalization, we let y 0 = 0, 0, f 1 s 0 1, s 0 2 = 0, f 3 s 0 2, s 0 3 = 0, and f2 s 0 1, s 0 2, s0 3 = 0. If the argaining etween Nodes 1 and 2 reak down, an event in the extensive-form argaining game to e defined, then Node 2 has a limited set of actions that can e taken with Node 3, and likewise with a reakdown etween Nodes 2 and 3. Specifically, in the event of a argaining reakdown etween Nodes 1 and 2, the action of Node 1 is its status quo action s 0 1, whereas the pair s 2, s 3 of actions of Nodes 2 and 3 must e chosen from S B 2,3 = { s 2, s 3 : s 2 S 1 2, s 3 C 3 s 2 }, where S 1 2 is a given non-empty suset of S 2 with the property that any action in S 1 2 is compatile with s 0 1. That is, s0 1 C 1s 2 for any s 2 S2 1. Likewise, in the event of a reakdown in the argaining etween Nodes 2 and 3, the action of Node 3 is its status quo action s 0 3, whereas the 5

7 actions of Nodes 1 and 2 must e chosen from S B 1,2 = { s 1, s 2 : s 2 S 3 2, s 1 C 1 s 2 }, where S 3 2 is a given non-empty suset of S 2 with the property that any action in S 3 2 are compatile with s 0 3. By assumption, SB 2,3 and SB 1,2 are not empty. We assume for simplicity that each stated argmax is a singleton that, is the associated maximization prolem has a unique solution. This is generically true in the space of utilities. 1 The socially optimal result is s 1, s 2, s 3 = argmax s 2 S 2 s 1 C 1 s 2 s 3 C 3 s 2 Us 1, s 2, s 3, 2 where U : S R is the social welfare function defined y Us = f 1 s 1, s 2 + f 2 s 1, s 2, s 3 + f 3 s 2, s 3. Our main result is a protocol for contingent pairwise argaining under which the unique extensive form tremling hand perfect equilirium, for any sufficiently small tremles, achieves the socially optimal actions s. We also provide alternative plausile argaining approaches that do not lead to this efficient result. 3 Simple Illustrative Example For an extremely simple illustrative example, we may imagine a situation in which firm 2 is negotiating credit agreements with firms 1 and 3. The creditor firms 1 and 3 each egin with 1 unit of cash at time zero. Firm 2 initially has c < 1 in cash, and has the opportunity to undertake a project that requires 2 units of cash. At time 1, the project will pay some amount A > 2 with success proaility p, and otherwise pays B, where 1 < B < 2. For some negotiated note discount y i, creditor firm i will provide firm 2 with 1 y i in cash at time zero in return for a note promising 1 That is, fixing all other primitives of the model, we can view the vectors of utilities of the firms or of a suset of firms defined y the utilities f 1, f 2, and f 3 as elements of a Euclidean space. A condition is said to hold generically in a Euclidean space if it holds except for a closed suset of zero Leesgue measure. 6

8 to pay 1 at time 1. Without loss of generality for this example, we can take S 1 = S 3 = {0} and C 1 s 2 = C 3 s 2 = {0}. Firm 2 chooses from S 2 = {0, 1} {0, 1} {1, 3}, each element of which specifies, respectively, whether credit is taken from 1, whether credit is taken from 3, and whether firm 1 or firm 3 receives seniority. If there is no agreement, all firms consume their initial cash. If firm 2 can negotiate funding from each of the creditors, then at time zero it will invest 2 in the risky project and consume all of its excess cash, which is c y 1 y 3. At time 1, firm 2 will consume A 2 if the risky project is successful and nothing otherwise. If the project is funded, then at time 0 firms 1 and 3 will consume y 1 and y 3 respectively. At time 1, these creditors will each consume 1 if the project is successful. Otherwise, the senior creditor will consume 1 and the junior creditor will consume B 1. Firm i has utility c 0 + γ i Ec 1 for consumption c 0 in period 0 and c 1 in period 1. We suppose that γ 2 < γ 1 < γ 3. The status-quo reakdown actions is taken to e s 0 2 = 0, 0, 3. Which of the creditors is senior in the event of no funding is irrelevant, and is taken to e firm 3 without loss of generality. The model s primitive set of parameters is thus A, B, p, c, γ 1, γ 2, γ 3, s 0 2. We suppose that the project is worth funding, in terms of total utility, no matter which creditor is senior. The unique efficient outcome, suject to the limited availale forms of credit agreements, is therefore to fund the project and for firm 3 to e the senior creditor. That is, s 2 = 1, 1, 3. We assume that y 1 + y 3 < c for any discounts y 1 and y 3 that are individually rational for firms 2. This is the condition c > γ 2 pa 2. With this, c is irrelevant and can e ignored when calculating an equilirium. We also assume that the set of discounts y 1, y 3 that are individually rational for all firms is not empty, even if the wrong creditor, firm 1, is senior. This is the condition γ 2 pa 2 > 2 γ 1 γ 3 p + 1 pb 1. After normalizing y sutracting the initial cash utility of 1, firms 1 and 3 receive nothing in the event that the project is not funded fully, and otherwise receive utility y i + f i 1, 1, j where f i 1, 1, i = γ i f i 1, 1, i = γ i p + 1 pb 1, 3 depending on whether firm i i otains seniority. We have suppressed from the notation the 7

9 dependence of f i on s 1 and s 3, since this is trivial. After normalizing y sutracting the initial cash utility of c, firm 2 receives nothing in the event that the project is not funded fully, and otherwise receives utility y 1 y 3 + f 2 1, 1, j = y 1 y 3 + γ 2 pa 2, regardless of which firm j otains seniority. 4 The Axiomatic Solution Appendix C provides foundations for an axiomatic solution of network ilateral argaining prolems. As we will show, the axiomatic solution coincides with the proposed equilirium for the associated non-cooperative extensive-form argaining game. In addition to i axioms for two-node networks that support the Bargaining Solution of Nash 1950, our axioms are ii multilateral staility, iii independence of irrelevant actions, and iv ilateral optimality. Under these axioms, we show that there is a uniquely defined and socially efficient solution, which we call the axiomatic solution. We riefly motivate the axiomatic solution here, and provide details in Appendix C. The main ojective of the paper is to show that the unique extensiveform tremling-hand-perfect equilirium for our non-cooperative extensive-form alternating-offers argaining game associated with contingent network contracting reaches the same payments and the same efficient actions as those uniquely specified y the axiomatic solution. This can e viewed as an extension to network games of the non-cooperative-game foundation estalished y Binmore, Ruinstein, and Wolinsky 1986 for Nash argaining. Formally, our axioms concern the properties of a solution F : Σ Ω, a function that maps the space Σ of network ilateral argaining prolems to the space Ω of associated actions and payments. These spaces Σ and Ω are formally defined in Appendix C. Our first axiom is that when applied to a network ilateral argaining prolem σ whose graph merely consists of two connected nodes, the solution F σ is effectively the Nash Bargaining Solution, 8

10 specifying actions s = s 1, s 2 and a payment y 12 that solve max 1s + f 2 s} s C 4 yij = 1 [fi s u 2 i ] [f j s u j ], 5 where u 1 and u 2 are the outside option values of nodes 1 and 2. In Appendix C, we discuss underlying axioms for two-player games that support this Nash Bargaining Solution. Roughly speaking, a solution F satisfies our second axiom, multilateral staility, if, for any given network ilateral argaining prolem σ, whenever one freezes the actions and payments among a suset of pairs of directly connected nodes, and then applies F to the argaining prolem σ su induced for remaining su-network G su, the solution F σ su of the su-network ilateral argaining prolem σ su coincides on the su-network with that prescried y F σ. Multilateral staility is illustrated in Figure 3. s 5 s 5 y 25 ỹ 25 s 2 s 2 y 12 y 24 ỹ 12 ỹ 24 s 1 y 23 s 4 s 1 ỹ 23 s 4 y 34 s 3 s 3 Figure 3: Multilateral staility. By freezing the argaining outcome of nodes 3 and 4 lue off the su-network formed y edges other than 3, 4, the solution on the induced su-network game coincides with the lue solution. That is, s i = s i and ỹ ij = y ij. Our third axiom, independence of irrelevant actions, states roughly that if the solution specifies some outcome s, y for a network ilateral argaining prolem, and if we alter this network ilateral argaining prolem merely y reducing the set of feasile actions while still admitting s 9

11 as feasile, then s, y remains the solution. A solution F satisfies the final axiom, ilateral optimality if, for any given network ilateral argaining prolem σ, when any two directly connected nodes maximize the sum of their total payoff under the assumption that the remainder of the nodes will react according the solution F applied to their su-network, then the maximizing actions they would choose are consistent with the solution F σ of the entire network ilateral argaining prolem. Bilateral optimality is illustrated in Figure 4. s 5 s 5 y 25 y 25 s 2 s 2 y 12 y 24 y 12 y 24 s 1 ỹ 23 s 4 s 1 y 23 s 4 y 34 y 34 s 3 s 3 Figure 4: Suppose the orange nodes, 2 and 3, maximize their total payoff assuming that the remaining network will react according to the lue solution. Bilateral optimality implies that the actions s 2, s 3 coincides with applying the lue solution to the whole network. That is, s 2, s 3 = s 2, s 3. A result stated in Appendix C implies that the axiomatic solution y a, s a associated with our our present three-firm network setting is given y the efficient actions s a = s and the upfront payments y a = y1 a, ya 3 that uniquely solve the equations u 1 y a, s u 0 12 = u 2 y a, s u 0 21 u 3 y a, s u 0 32 = u 2 y a, s u 0 23, 6 10

12 where u 0 ij is the outside-option value of node i in its argaining with node j. Here u 0 12 = u 0 32 = 0 u 0 21 = 1 2 u 0 23 = 1 2 max s 2,s 3 S B 2,3 max s 1,s 2 S B 1,2 U s 0 1, s 2, s 3 7 U s 1, s 2, s 0 3. The main result of the paper, stated in Section 8, provides simple conditions under which this axiomatic solution is also the unique extensive-form tremling-hand equilirium of the non-cooperative contingent-contract ilateral argaining game to e descried. In this sense, ilateral argaining over complete contingent contracts is socially efficient in our network market setting. It also follows that our non-cooperative equilirium solution concept for extensive-form ilateral argaining over contingent contracts satisfies multilateral staility, irrelevance of independent actions, and ilateral optimality. 5 Counterexamples This section explores variants of the model definition or solution concept that do not lead to efficient equilirium outcomes. Our ojective here is to promote an understanding of the dependence of our main efficiency results on our assumptions. For this purpose, we will restrict attention to the simple special case in which f 1 = 0, f 3 = 0, and S 2 = { s 0 2}. That is, utility is otained only y the central node, and the treatment of the central node is fixed. We will simply write f for f Bargaining Without Communication We will first consider ilateral argaining without the aility of either pair of connected nodes to contract on the argaining outcome of the other pair. That is to say, when Nodes 1 and 2 argain, Nodes 1 and 2 are unale write a contract that depends on the action s 3 of Node 3 or the payment y 3. Likewise, when Nodes 2 and 3 argain, Node 3 cannot contract on the action s 1 of Node 1 or the payment y 1. The ehavior of Nodes 1 and 3 in such a game depends on their eliefs aout the contracted action and payment of other nodes. 11

13 In this setting, it is common to restrict attention to passive eliefs in which, after oserving a deviation, each node continues to elieve that other nodes receive their equilirium offers. This is typical in Hart, Tirole, Carlton, and Williamson 1990 and Segal Let s 1, s 3 ; y 1, y 3 denote the equilirium outcome. With passive eliefs, if Node i is offered s i, y i s i, y i, he still elieves that other nodes make their equilirium choices of treatments and prices. We suppose that the outcomes of ilateral negotiations are given y the Nash Bargaining Solution NBS. The passiveeliefs equilirium must therefore e a pairwise stale Nash argaining solution. That is, s i, y i is the Nash solution to the argaining prolem etween Node i and Node 2, under the elief that s j, y j is the agreed choice y Node j and Node 2. Hence s, y solves 2 max y 1 [fs 1, s 3 y3 y 1 ] s 1 S 1,y 1 R max y 3 [fs 1, s 3 y1 y 3 ]. s 3 S 3,y 3 R 8 From 8, it is straightforward to characterize pairwise-stale NBS y s 1 = argmax s1 S 1 fs 1, s 3 s 3 = argmax s3 S 3 fs 1, s 3 y 1 = y 3 = 1 3 fs 1, s 3. In a pairwise stale NBS, each node has the same utility u 1 = u 2 = u 3 = 1 3 fs 1, s 3. We can see that the efficient vector of treatments s is indeed consistent with a pairwise stale NBS. Not every pairwise stale NBS, however, is necessarily efficient. This is so ecause s 1, s 3 2 One may argue that B i = {fs i, s j y j y i, y i : s i S i, y i R} is not convex, whereas the Nash solution requires convexity. Indeed, the payoff pairs form a finite numer of parallel lines in the Euclidean plane. One can convexify this set y filling in the gaps etween the lines. Then the axiom of Independence of Irrelevant Alternatives implies that the unique solution is given y maximizing the utility products in equation 8. 12

14 merely solves s 1 argmax s 1 S 1 fs 1, s 3 s 3 argmax s 3 S 3 fs 1, s 3, 9 whereas s 1, s 3 jointly maximizes fs 1, s 3. Depending on the utility function f, there may e other pairwise stale NBS, which are Pareto ranked. In a pairwise stale NBS, Nodes 1 and 3 cannot e sure of making efficient choices ecause of the inaility to contract ased on communication etween the two spoke-end nodes in the solution concept. Node i cannot e certain that node j will choose the efficient treatment s j. Suppose a pair of treatments s 1, s 3 satisfies 9. If Node i elieves that Node j chooses s j, then the outcome of her argaining with the central node is s i. One can construct a non-cooperative alternating-offers game whose Perfect Bayesian Nash equiliria with passive eliefs coincide with the pairwise stale NBS, in the limit as players ecome infinitely patient. This is shown y de Fontenay and Gans In such a game, the central node is assumed to e ale to argain only over prices, ut not the actions to e chosen y the spoke-end nodes. Thus the central node cannot efficiently coordinate the actions of the spoke ends. In practice, a orrowing firm is typically ale to credily assign higher seniority to one lender over another, in return for a low interest rate. A junior lender is ale, through accounting disclosure or covenants, is typically ale to receive information on its relative and asolute loss of priority and demand a correspondingly high interest rate. In other cases, however, firms in a network may fail to coordinate their contracts and uses price negotiations to promote efficient outcomes due to a lack verifiale and contractile information. 5.2 Distortion of Outside Option Values We now allow Nodes 1 and 2 to sign enforceale contingent contracts. For each action s 3 chosen y Node 3, Nodes 1 and 2 can choose a different action-payment pair s 1, y 1. Nodes 2 and 3 sign a simple, non-contingent inding contract s 3, y 3, to which oth nodes commit. This setting is equivalent to the following 2-stage game: Nodes 2 and 3 first argain over s 3, y 3 in Stage 1. Then, in Stage 2, Node 2 argains with Node 1 over s 1 s 3, y 1 s 3, with common knowledge of the action 13

15 s 3 chosen y Node 3. We show that the socially efficient outcome s 1, s 3 may not e an equilirium. Suppose in Stage 1, Nodes 2 and 3 choose s 3, y 3. Then, in Stage 2, a reakdown etween Nodes 1 and 2 leads to payoffs of u 0 12 s 3 = 0 for Node 1 and u 0 21 s 3 = f s 0, s 3 y3 for Node 2. Therefore u 0 12 s 3 and u 0 21 s 3 are the respective outside option values for Nodes 1 and 2 in their ilateral argaining in Stage 2. Likewise, the outside option values for Nodes 2 and 3 in Stage 1 are u 0 23 = 0.5f s 1 s 0 3 and u 0 32 = 0, where s 1 s 3 = argmax s1 S 1 fs 1, s 3. By the same argument used to determine the pairwise stale NBS, the Nash argaining outcome s 1 s 3, y 1 s 3 etween Nodes 1 and 2 in Stage 2 is s 1s 3 = argmax s 1 S 1 fs 1, s 3, whereas the payment y 1 s 3 is determined y y 1 s 3 = fs 1 s 3, s 3 y 1 s 3 y 3 u 0 21 s 3 y 3 = fs 1 s 3, s 3 y 1 s 3 y 3 u Hence y 1 s 3 = 1 2 [ fs 1 s 3, s 3 f s 0 1, s ] 3, 2y 3 = 1 2 [ fs 1 s 3, s 3 + f s 0 1, s 3] 1 2 f s 1 s 0 3, s 0 3. Therefore, in Stage 2, when choosing s 3, Nodes 2 and 3 s 3 oth receive the payment gs 3 = 1 2 [ 1 fs 2 1 s 3, s 3 + f s 0 ] 1 1, s 3 2 f s 1 s 0 3, s 0 3. One can easily choose f so that the maximum of g is not attained at s 3 = s 3. In this case, the socially efficient outcome s 1, s 3 cannot e an equilirium. Indeed, y committing to s 3, when argaining with Node 1, Node 2 has an outside option value u 0 21 s 3 = f s 0, s 3 y3 that depends on s 3. In this sense, the choice of s 3 in Stage 1 may distort the outside option value of Node 2 in Stage 2. A low value of u 0 21 forces Node 2 to make a high payment to Node 1, which is detrimental to oth Nodes 2 and 3. Thus the distortion caused y 14

16 this outside option value u 0 21 can create an incentive for inefficient equilirium outcomes. 5.3 Incentives to Lie Distort the Distriution of Surplus We again allow nodes to sign contingent contracts. For each action s 3 chosen y Node 3, Nodes 1 and 2 choose some action-payment pair s 1, y 1, and vice versa. We now assume, however, that the contingent contracts are not enforceale. We will see, not surprisingly, that the central node may have an incentive to misreport to one end node the outcome of its argain with the other end node. We assume that the outside option values for oth ilateral argaining prolems that etween Nodes 1 and 2, and that etween Nodes 2 and 3 are all 0. This assumption allows us to isolate the effect of dishonesty y the central node. As this rules out the distortion of outside option values through commitment to a contract, we consider only inding contracts. Consider the following 2-stage game. In Stage 1, Nodes 2 and 3 first argain over s 3, y 3. In Stage 2, Node 2 communicates to Node 1, not necessarily truthfully, that the action chosen y Node 3 is s 3. Then Nodes 1 and 2 argain over s 1 s 3, y 1 s 3. Truthful communication y Node 2 and the socially efficient outcome s 1, s 3 need not e an equilirium of the game. In order to see this, suppose this is in fact an equilirium. In Stage 1, Nodes 2 and 3 choose the treatment s 3 along with some payment y 3 R. If Node 1 elieves the report from Node 2 that Node 3 agreed to take the action s 3, then following the earlier determination of pairwise stale NBS, the Nash argaining outcome s 1 s 3, y 1 s 3 etween Nodes 1 and 2 in Stage 2 would e: s 1 s 3 = argmax s 1 S 1 fs 1, s 3. The associated payment y 1 s 3 is determined y y 1 s 3 = fs 1 s 3, s 3 y 1 s 3 y 3 y 3 = fs 1 s 3, s 3 y 1 s 3 y 3. 15

17 Hence y 1 s 3 = 1 3 fs 1 s 3, s 3. The value to Node 2 associated with reporting s 3 is f s 1 s 3, s 3 y 1 s 3 y 3 = f s 1 s 3, s fs 1 s 3, s 3 y 3. We define g : S 3 R y g s 3 = f s 1 s 3, s fs 1 s 3, s 3. One can choose f so that the maximum of g is not attained at s 3 = s 3. It is therefore not credile that Node 2 correctly reports. This could destroy the socially efficient outcome s an equilirium. 1, s 3 for eing We have shown that if Node 1 elieves the report of Node 2, then Node 2 will not report truthfully, so there is no truth-telling equilirium. There are other possiilities, ased on a definition of equilirium in which Node 1 does not necessarily elieve Node 2, ut rather makes an inference aout s 3 ased on the report s 3. In this setting, it is conceivale that an equilirium may not exist, or that there may e either efficient or inefficient equiliria. We intend to go further into this in the next version of the paper. 6 Contingent-Contract Network Bargaining Our main ojective now is to explore a particular argaining protocol, extending the approach of Ruinstein 1982 and Binmore, Ruinstein, and Wolinsky 1986, y which the limit of the unique tremling-hand-perfect equiliria of the corresponding extensive-form negotiation games, as reakdown proailities converge to zero, is the axiomatic solution y a, s a, and in particular achieves the socially efficient choice s a = s. As explained in the previous section, not all plausile extensions of the Ruinstein model to our network setting have this efficiency property. We allow the actions negotiated y Nodes 1 and 2 to e contractually contingent on the actions to e chosen y Nodes 2 and 3 from S 2,3 = {s 2, s 3 : s 2 S 2, s 3 C 3 s 2 }. As we shall see, Nodes 2 and 3 may experience a reakdown in their negotiation, a contingency that we lael B 2,3. Thus the set of possile outcomes of the argaining of Nodes 2 and 3 is S 2,3 {B 2,3 }. The contingent 16

18 action to e negotiated etween Nodes 1 and 2 is s 1, s 1, s1 2, where s1 is from the space C = {s 1 : S 2,3 S 1 : s 1 s 2, s 3 C 1 s 2 }, and s 1, s1 2 is chosen from S B 1,2. One should understand s 1, s1 to e the pair of actions of Nodes 1 and 2 that apply if Nodes 2 and 3 reak down. Thus, the contract etween Nodes 1 and 2 2 will e expressed in the form of some contingent payment y 1 : S 2,3 R, or y 1 R in the event B 2,3 of a reakdown in argaining etween Nodes 2 and 3, and some contingent action s 1 in C, or s 1, s 1 2 S B 1,2 in the same reakdown event B 2,3. The contract to e chosen y Nodes 2 and 3 is some s 2, s 3 S 2,3 along with some associated payment y 3, and some s 3 2, s 3 S B 2,3 in the event of a reakdown etween Nodes 1 and 2, along with some associated payment y 3 R. The proposed extensive-form network argaining game is defined as follows, in four stages. Stage a: In Stage a, Nodes 1 and 2 argain over their contingent action. We suppose that Node 1 is the first proposer and offers a contingent action s 1 in C and some s 1, s1 2 S B 1,2. The identity of the first proposer is irrelevant for the ultimate solution concept. For each contingency s 2, s 3 S 2,3, Node 2 either accepts or rejects s 1 s 2, s 3. Likewise, for the contingency B 2,3 of a reakdown etween Nodes 2 and 3, Node 2 either accepts or rejects s 1, s1 2. Acceptance closes the argaining etween 1 and 2 over the action s 1 respectively, s 1, s1 2 contingent on s 2, s 3 respectively, on B 2,3. Agreement or rejection at one contingency does not ind ehavior at any other contingency. Rejection at a particular contingency including B 2,3 leads, with a given proaility η, to a reakdown of the negotiation over that contingency. If Nodes 1 and 2 reak down over s 1, s1 2, the resulting actions for oth Nodes 1 and 2 are the exogenous statusquo choices, s 1, s1 2 = s0 1, s0 2. These reakdown events are independent across contingencies. The process continues to the next period, when Node 2 is the proposer and Node 1 accepts or rejects, as illustrated in Figure 5. This alternating-offers procedure is iterated until agreement or reakdown. Let S t 2,3 e the set of contingencies including B 2,3 that are still open for negotiation. That is, S2,3 t is the set of contingencies for which Nodes 1 and 2 have reached neither agreement nor reakdown y the eginning of period t. In period t, Nodes 1 and 2 argain over s 1 s 2, s 3 for these remaining contingencies s 2, s 3 in S2,3 t, and over s 1, s1 2 for the contingency B2,3, if B 2,3 S2,3 t. 17

19 agreement accept s propose s firm 1 firm 2 reject 1 η propose s firm 2 firm 1 nature η reakdown s 0 Figure 5: The first four stages of a generic Ruinstein alternating-offers game. The argaining etween Nodes 1 and 2 concludes at the first time y which they have reached an agreement or roken down for all of contingencies in S 2,3 {B 2,3 }. This is a finite time, almost surely. The reakdown proaility η is an exogenous parameter of the model. We will later e interested in the limit ehavior as η 0. The result of Stage a of the argaining game is some random set X 2,3 S 2,3 {B 2,3 } on which Nodes 1 and 2 reach agreement, and for each contingency in X 2,3, the agreed action s 1 s 2, s 3 for Node 1, as well as the agreed pair of actions s 1, s1 2 in the contingency B 2,3, if it is in X 2,3. Stage : In Stage, Nodes 2 and 3 argain without contingencies over some s 2, s 3, s 3 2, 3 s such that s 2, s 3 S 2,3 and s 3 2, s 3 S B 2,3. One should understand s 3 2, 3 s to e the actions of Nodes 2 and 3 that apply in the event that Nodes 1 and 2 reak down at the contingency s 2, s 3. Simultaneous with the argaining in Stage a, Nodes 2 and 3 play a similar alternating-offers argaining game. We suppose that Node 3 proposes first. Again, the identity of the first proposer does not matter in the limit as η 0. The two negotiations, etween Nodes 1 and 2 in Stage a, and etween Nodes 2 and 3 in Stage, are not coordinated in any way. Specifically, Stage actions cannot depend on information from ongoing play or reakdowns in Stage a, and vice versa. Let A 2,3 {Y, N} e the inary variale indicating whether Node 2 and 3 reach an agreement Y or not N over s 2, s 3. If Nodes 2 and 3 reak down over s 3 2, s 3, the resulting actions for oth Node 2 and 3 are the 18

20 exogenous status-quo choices, s 3 2, s 3 = s 0 2, s 0 3. Stage aa: Once Nodes 1 and 2 have finished Stage a and oserved the resulting set X 2,3 of contingencies at which they agree on an action s 1 s 2, s 3 and actions s 1, s1 2 for B2,3 if B 2,3 X 2,3, they argain in Stage aa over the corresponding payments y 1 s 2, s 3 and y 1, respectively. If B 2,3 X 2,3, that is, if Nodes 1 and 2 did not reach an agreement over s 1, s1 2 in Stage a, then y 1 is the null payment y1 = 0. Otherwise when argaining over the payments y 1s 2, s 3 and y1, Nodes 1 and 2 use the same form of alternating-offer game. For any s 2, s 3 X 2,3 respectively, for B 2,3, reakdown of the payment argaining leads to the status-quo action s 0 1 respectively, actions s 0 1, s0 2 and the null payment, y1 0 = 0. That is, unless they can agree on the payment y 1s 2, s 3 respectively, y1, the contingent action s 1 s 2, s 3 respectively, actions s 1, s1 2 agreed in Stage a is discarded. We let X2,3 t e the set of contingencies that are still open for negotiation at the eginning of period t in Stage aa. The result of Stage aa is a random set A 2,3 X 2,3 at which there is ultimately agreement on oth an action s 1 s 2, s 3 and a payment y 1 s 2, s 3, as well as actions s 1, s1 2 and a payment y 1 for the contingency B 2,3. Stage : Similarly, following Stage, Nodes 2 and 3 argain over the payment y 3 associated with s 2, s 3 and the payment y 3 associated with s 3 2, s 3 to e made in the event that Nodes 1 and 2 do not reach agreement at the choice s 2, s 3 of Nodes 2 and 3. As with the paired Stages a and, the negotiations and reakdowns in Stages aa and are carried out independently. In summary, given the results s 1, s 1, s1 2 ; y 1, y1, A 2,3 and s2, s 3, s 3 2, s 3 ; y 3, y3, A 2,3 of the four stages, the actions and payments of the game are determined as follows. If A 2,3 = Y and s 2, s 3 A 2,3, then the ultimate actions and payments are [s 1 s 2, s 3, s 2, s 3 ; y 1 s 2, s 3, y 3 s 2, s 3 ]. If, instead, A 2,3 = Y and s 2, s 3 A 2,3, the outcome of the argaining game is [ s 0 1, s3 2, s 3 ; y0 1 = 0, y 3]. Finally, in the event that A 2,3 = N, the final outcome is [ s 1, s1 2, s0 3 ; y 1, y0 3 = 0]. This comination of the four stages into final results is illustrated in Figure 6. An extensive-form argaining game is defined in this manner for each list η, S, C, f, s 0 of model parameters, where S = S 1, S 2, S 3, S 1 2, S3 2, C = C1, C 3, f = f 1, f 2, f 3, and s 0 = s 0 1, s0 2, s0 3. We could have merged Stages a and aa and likewise have merged Stages and without strategic difference. We split the game into these stages, however, in order to take advantage of the 19

21 Final result [s 1s 2, s 3, s 2, s 3; y 1s 2, s 3, y 3s 2, s 3] if A 2,3 = Y and s 2, s 3 A 2,3 s 0 1, s 3 2, s 3; y1, 0 y3 if A 2,3 = Y and s 2, s 3 A 2,3 s 1, s 1 2, s 0 3; y1, y3 0 if A2,3 = N Nodes 1 and 2 in Stage aa {y 1s 2, s 3 : s 2, s 3 A 2,3}, y 1 Nodes 2 and 3 in Stage y3, y 3 Nodes 1 and 2 in Stage a {s 1s 2, s 3 : s 2, s 3 X 2,3}, s 1, s 1 2 Nodes 2 and 3 in Stage s2, s 3; s 3 2, s 3 Figure 6: The stages of the ilateral argaining encounters refinement associated with extensive form tremling-hand perfection, which we turn to next. As we explain at the end of Section 8, a failure to split the game into stages would admit additional weird and inefficient equiliria that survive the tremling-hand perfection refinement. 7 Extensive Form Tremling-Hand Perfection Our next ojective is to select what we elieve to e a natural equilirium for our OTC market game, using the tremling-hand refinement concept of Selten For an extensive-form game such as ours, the tremling-hand refinement is ased on the agent-normal form of the game, which in our setting involves a countaly infinite numer of players ecause of the aritrarily many rounds of alternating offers. Further, each monetary compensation y i is chosen from a continuum set, so our extension must also e ased on somewhat general topological strategy spaces. This section is devoted to the necessary extension of standard results for tremling-hand-perfect equilirium. Although these results may e new, they are not surprising, and any reader familiar with tremling- 20

22 hand refinements for finite-dimensional games could safely skip this section without much loss. We first provide some asic definitions. Let N denote a set of players that is finite or countaly infinite. Each player i N has a pure-strategy set X i and a payoff function u i : X R, where X = i N X i. The associated game is denoted G = X i, u i i N. A strategy profile x X is defined to e a pure-strategy Nash equilirium of G if, for each i N, u i x u i x i, x i, for every x i X i, where x i, x i X is constructed as usual from a given x in X y replacing the pure-strategy x i of player i with x i. We suppose that each X i is a measurale space and that each u i is ounded and measurale with respect to the product σ-algera on the product space X. Letting M i e the set of proaility measures on X i, we extend u i to M = i N M i y defining u i µ = u i x dµx. X The existence and uniqueness of the product measure µ is ensured y Theorem 20 of Dunford and Schwartz , p We call G = M i, u i i N the mixed extension of G. Given a game G, a Nash equilirium of the mixed extension G is called a mixed-strategy Nash equilirium of G. By a slight ause of terminology, we sometimes refer to a mixed-strategy Nash equilirium of G simply as a Nash equilirium of G. We now further assume that each X i is a topological space. In this case we call G a topological game. We endow X i with the Borel σ-algera and let M i e the set of Borel proaility measures on X i, endowed with the weak* topology. Such a measure µ i is said to e strictly positive if µ i O > 0 for every nonempty open set O in X i. For each i, we let M i M i e the set of all such strictly positive measures, and set M = M i N i. For ν M and δ [0, 1, we define M i δν i = {µ i M i : µ i δν i } and Mδν = i N M iδν i. We will consider restricting players to the mixed strategies in Mδν, and in that case let u i Mδν denote the associated restriction of the utility function for player i. 21

23 The game G δν = M i δν i, u i Mδν i N is called a Selten perturation of G. The following definition extends the solution concept of tremling-hand perfection, in terms of Selten perturations, to games with topological strategy spaces and countaly many players. Definition 1. A strategy profile µ M is a tremling-hand perfect thp equilirium of G if there is a sequence δ n, ν n, µ n, with δ n 0, 1, ν n M, and µ n Mδν with µ n µ, such that δ n 0 and each µ n is a Nash equilirium of the pertured game G δ n ν n. As usual, µn µ denotes weak* convergence. Roughly speaking, µ is a thp equilirium of G if it is the limit of some sequence of equiliria of perturations of G that are closer and closer to the target game G. The following definition extends the perfection concept of Simon and Stinchcome 1995 to games with topological strategy spaces and with countale many players. For a given strategy vector µ M, the set of est responses of player i is Br i µ = { σ i M i : u i σ i, µ i sup ρ i M i ρ i, µ i }. The total-variation distance etween two proaility measures P and Q on a measurale space Ω, F is d T V P, Q = sup P A QA. A F Definition 2. Given a real ɛ > 0, some µ M is a strong ɛ-perfect equilirium of G if, for each i N, d T V µ i, Br i µ ɛ. A strategy profile µ M is a strong perfect equilirium of G if it is the weak* limit of a sequence of strong ɛ n -perfect equiliria, for some ɛ n 0. The following result estalishes the relationship etween tremling-hand perfection and strong perfection. The equivalence of 1-3 is analogous to standard characterizations of tremling-hand 22

24 perfect equiliria in games with finite strategy sets and finitely many players, as in Fudenerg and Tirole Proposition 1. For a topological game G = X i, u i i N, the following three conditions are equivalent: 1 µ is a tremling-hand perfect equilirium of G. 2 µ is a strong perfect equilirium of G. 3 µ is the limit of a sequence µ n in M with the property that for every ɛ > 0, for sufficiently large n, and for each i, µ n i { x i X i : u i x i, µ n i sup u i x i, µ n i x i X i } 1 ɛ. When writing x i, µ n i, we take x i to e the dirac measure at x i. We also note that, ecause u i } is measurale, {x i X i : u i x i, µ n i sup x i u ix Xi i, µn i is indeed a Borel set, as needed. For a topological game G = X i, u i i N, let T G e the set of thp equiliria of G. For any µ T G and any non-negative real sequence ɛ n 0, let T µ; ɛ n n 0 e the set of sequences of ɛ n -perfect equiliria converging to µ. Finally, let T µ = {ɛ n : ɛ n 0} T µ; ɛn n 0. Proposition 2. For a topological game G = X i, u i i N, suppose µ T G is a thp equilirium of G, and µ n T µ; ɛ n n 0 for some sequence ɛ n 0. Then for each player i there exists µ n i Br i µ n such that µ n i µ i. Proof. This is immediate from the definition of strong perfect equilirium and the following fact: If P n and Q n are sequences of Borel proaility measures on some topological space X such that, d T V P n, Q n 0 and P n P for some Borel proaility measure P on X, then Q n P. Normal form tremling-hand-perfect equiliria need not e sugame perfect, even in a finite game. One can demonstrate this in two-player games, where normal form tremling hand perfection is equivalent to selecting Nash equiliria in which no player chooses a weakly dominated strategy. Avoiding weakly dominated strategies in the normal form is not sufficient to guarantee sugame perfection. 23

25 The agent normal form of an extensive-form game is the normal-form game that one would otain if each player selected a different agent to make her decisions at every information set, and if all of the original players agents act independently with the oject of maximizing the original player s payoff. Definition 3. A strategy profile is extensive form tremling-hand perfect EFTHP if it is normalform tremling-hand perfect in the agent normal form of the same game. In a finite game, an EFTHP equilirium is sugame perfect. 8 Solving for EFTHP Equiliria The game descried in Section 6 is an extensive form game with imperfect information. We look for strategy profiles that are extensive-form tremling-hand perfect. For this, we first specify a topology on each strategy space. 8.1 Strategy Spaces Let X i,t,k denote the strategy space of node i at period t in stage k {a,, aa, }. Because of agent perfection, the suscript i, t, k also refers to the identity of the agent who represents node i at period 0 in Stage k. For example, X 10a is the strategy space of agent 10a. The strategy spaces of Stages a and are finite and endowed with the discrete topology in which each singleton is an open set. A totally mixed strategy on X i,t,a or X i,t, is thus a strategy that assigns strictly positive proaility to each pure strategy. As for the infinite strategy sets X itaa and X it, we first fix a compact interval K = [ M, M] R such that max s2 S 2, s 1 C 1 s 2 f 1 s 1, s 2 M 2 max s2 S 2, s 3 C 3 s 2 f 3 s 2, s 3 M 2 max s2 S 2, s 1 C 1 s 2, s 3 C 3 s 2 f 2 s 1, s 2, s 3 M 2 < min s2 S 1 2 f 1 s 0 1, s 2, < min s2 S 3 2 f 3 s2, s 0 3, < f 2 s 0 1, s 0 2, s0 3 = 0. 24

26 Because any payment greater than M is not individually rational for at least one of the nodes, we can without loss of generality restrict payments to the space K. We now specify X itaa. The specification of X it is similar and omitted. Let H t,aa e the set of all potential histories of play up to time t in Stage aa. A strategy for agent itaa is a function assigning to each history in H t,aa an action in the relevant strategy space. When t is even, the strategy space of 1taa is X 1taa = K X ht,aa, h t,aa H t,aa where X h t,aa is the set of contingencies that are still open for negotiation at the eginning of period t under history h t,aa. That is, 1taa should offer a payment in K for each contingency in X h t,aa. We write x 1taa h t,aa ; s 2, s 3 for x 1taa h t,aa s 2, s 3. We endow X 1taa with the product topology. By Tychonoff s Theorem, X 1taa is a compact Hausdorff space. When t is odd, 1taa decides which offers to accept and which to reject, for each given history h t,aa. We denote acceptance y 1 and rejection y 0. Thus X 1taa = h t,aa H t,aa K X ht,aa, where K = {0, 1} K. Again, when t is odd, X 1taa is compact and Hausdorff. A compact Hausdorff strategy space for agent 2taa is similarly defined. This is preliminary. In a future revision, we will revisit these definitions in order to address measuraility concerns. 8.2 Equilirium Strategies We now define candidate equilirium strategies, eginning with some notation. The candidate equilirium contingent action s 1 in C is defined y s 1s 2, s 3 = argmax f 1 s 1, s 2 + f 2 s 1, s 2, s s 1 C 1 s 2 25