Showing papers on "Stochastic game published in 2003"

The Nonstochastic Multiarmed Bandit Problem

Abstract: In the multiarmed bandit problem, a gambler must decide which arm of K nonidentical slot machines to play in a sequence of trials so as to maximize his reward. This classical problem has received much attention because of the simple model it provides of the trade-off between exploration (trying out each arm to find the best one) and exploitation (playing the arm believed to give the best payoff). Past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines. In this work, we make no statistical assumptions whatsoever about the nature of the process generating the payoffs of the slot machines. We give a solution to the bandit problem in which an adversary, rather than a well-behaved stochastic process, has complete control over the payoffs. In a sequence of T plays, we prove that the per-round payoff of our algorithm approaches that of the best arm at the rate O(T-1/2). We show by a matching lower bound that this is the best possible. We also prove that our algorithm approaches the per-round payoff of any set of strategies at a similar rate: if the best strategy is chosen from a pool of N strategies, then our algorithm approaches the per-round payoff of the strategy at the rate O((log N1/2 T-1/2). Finally, we apply our results to the problem of playing an unknown repeated matrix game. We show that our algorithm approaches the minimax payoff of the unknown game at the rate O(T-1/2).

Global Games: Theory and Applications

Abstract: Many economic problems are naturally modeled as a game of incomplete information, where a player’s payoff depends on his own action, the actions of others, and some unknown economic fundamentals. For example, many accounts of currency attacks, bank runs, and liquidity crises give a central role to players’ uncertainty about other players’ actions. Because other players’ actions in such situations are motivated by their beliefs, the decision maker must take account of the beliefs held by other players. We know from the classic contribution of Harsanyi (1967–1968) that rational behavior in such environments not only depends on economic agents’ beliefs about economic fundamentals, but also depends on beliefs of higher-order – i.e., players’ beliefs about other players’ beliefs, players’ beliefs about other players’ beliefs about other players’ beliefs, and so on. Indeed, Mertens and Zamir (1985) have shown how one can give a complete description of the “type” of a player in an incomplete information game in terms of a full hierarchy of beliefs at all levels. In principle, optimal strategic behavior should be analyzed in the space of all possible infinite hierarchies of beliefs; however, such analysis is highly complex for players and analysts alike and is likely to prove intractable in general. It is therefore useful to identify strategic environments with incomplete information that are rich enough to capture the important role of higher-order beliefs in economic settings, but simple enough to allow tractable analysis. Global games, first studied by Carlsson and van Damme (1993a), represent one such environment. Uncertain economic fundamentals are summarized by a state θ and each player observes a different signal of the state with a small amount of noise. Assuming that the noise technology is common knowledge among the players, each player’s signal generates beliefs about fundamentals, beliefs about other players’ beliefs about fundamentals, and so on. Our purpose in this paper is to describe how such models work, how global game reasoning can be applied to economic problems, and how this analysis relates to more general analysis of higher-order beliefs in strategic settings.

Playing large games using simple strategies

Abstract: We prove the existence of e-Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be e-approximated by the payoffs to the players in some such logarithmic support e-Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasi-polynomial algorithm for computing an e-Nash equilibrium. To our knowledge this is the first subexponential algorithm for finding an e-Nash equilibrium. Our results hold for any multiple-player game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a fixed number of players m, the payoffs to all players in any m-tuple of mixed strategies can be e-approximated by the payoffs in some m-tuple of constant support strategies.We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an e-equilibrium with small support. It also implies that if the payoff matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.

Endogenous Formation of Links Between Players and of Coalitions: An Application of the Shapley Value

Abstract: Consider the coalitional game v on the player set (1,2,3) defined by $$ v(S) = \left\{ \begin{array}{l} 0\quad if{\kern 1pt} \left| S \right| = 1, \\ 60\quad if{\kern 1pt} \left| S \right| = 2, \\ 72\quad if{\kern 1pt} \left| S \right| = 3, \\ \end{array} \right. $$ (1) were |S| denotes the number of players in S. Most cooperative solution concepts “predict” (or assume) that the all-player coalition {1, 2, 3} will form and divide the payoff 72 in some appropriate way. Now suppose that P 1 (player 1) and P 2 happen to meet each other in the absence of P3. There is little doubt that they would quickly seize the opportunity to form the coalition {1, 2} and collect a payoff of 30 each. This would happen in spite of its inefficiency. The reason is that if Pi and P 2 were to invite P 3 to join the negotiations, then the three players would find themselves in effectively symmetric roles, and the expected outcome would be {24,24,24}. P1 and P 2 would not want to risk offering, say, 4 to P 3 (and dividing the remaining 68 among themselves), because they would realize that once P 3 is invited to participate in the negotiations, the situation turns “wide open” — anything can happen.

Strategic Experimentation with Exponential Bandits

Abstract: This paper studies a game of strategic experimentation with two-armed bandits whose risky arm might yield a payoff only after some exponentially distributed random time. Because of free-riding, there is an inefficiently low level of experimentation in any equilibrium where the players use stationary Markovian strategies with posterior beliefs as the state variable. After characterizing the unique symmetric Markovian equilibrium of the game, which is in mixed strategies, we construct a variety of pure-strategy equilibria. There is no equilibrium where all players use simple cut-off strategies. Equilibria where players switch finitely often between the roles of experimenter and free-rider all lead to the same pattern of information acquisition; the efficiency of these equilibria depends on the way players share the burden of experimentation among them. In equilibria where players switch roles infinitely often, they can acquire an approximately efficient amount of information, but the rate at which it is acquired still remains inefficient; moreover, the expected payoff of an experimenter exhibits the novel feature that it rises as players become more pessimistic. Finally, over the range of beliefs where players use both arms a positive fraction of the time, the symmetric equilibrium is dominated by any asymmetric one in terms of aggregate payoffs.

Trust, Risk and Betrayal

Abstract: Using experiments, we examine whether the decision to trust a stranger in a one-shot interaction is equivalent to taking a risky bet, or if a trust decision entails an additional risk premium to balance the costs of trust betrayal. We compare a binary-choice Trust game with a structurally identical, binary-choice Risky Dictator game with good or bad outcomes. We elicit individuals' minimum acceptable probabilities (MAPs) of getting the good outcome such that they would prefer the chance to the sure payoff. First movers state higher MAPs in the Trust game than in situations where nature determines the outcome.

Truth or Consequences: An Experiment

Abstract: This paper presents evidence that the willingness to punish an unfair action is sensitive to whether this action was preceded by a deceptive message. One player first sends a message indicating an intended play, which is either favorable or unfavorable to the other player in the game. After the message, the sender and the receiver play a simultaneous 2 x 2 game, in which the sender may or may not play according to his message. Outcome cells may, hence, be reached following true or false messages. In the third stage, the receiver may (at a cost) punish or reward, depending on which cell of the simultaneous game has been reached. We test whether receivers' rates of monetary sacrifice depend on the process by which an outcome is reached. We study two decision-elicitation methods: the strategy and the direct response methods. For each method, deception more than doubles the punishment rate as a response to an action that is unfavorable to the receiver. We also find evidence that 17--25% of all participants choose to reward a favorable action choice made by the sender, even though doing so leaves one at a payoff disadvantage. Our results reflect on current economic models of utility and have implications for organizational decision-making behavior.

Network Formation and Social Coordination

Abstract: This paper develops a simple model to examine the interaction between partner choice and individual behavior in games of coordination. An important ingredient of our approach is the way we model partner choice: We suppose that a player can establish ties with other players by unilaterally investing in costly pair-wise links. In this context, individual efforts to balance the costs and benefits of links are shown to lead to a unique equilibrium interaction architecture. The dynamics of network formation, however, has powerful effects on individual behavior: If costs of forming links are below a certain threshold then players coordinate on the risk-dominant action, while if costs are above this threshold then they coordinate on the efficient action. These findings are robust to a variety of modifications in the link formation process. For example, it may be posited that, in order for a link to materialize, the link proposal must be two-sided (i.e. put forward by both agents); or that, in case of a unilateral proposal, the link may be refused by the other party (if, say, the latter's net payoff is negative); or that a pair of agents can play the game even if connected only through indirect links.

Discounting the future in systems theory

Abstract: Discounting the future means that the value, today, of a unit payoffis 1 if the payoff occurs today, a if it occurs tomorrow, a2 if it occurs the day after tomorrow, and so on, for some real-valued discount factor 0 ≤ a ≤ 1. Discounting (or inflation) is a key paradigm in economics and has been studied in Markov decision processes as well as game theory. We submit that discounting also has a natural place in systems engineering: for nonterminating systems, a potential bug in the faraway future is less troubling than a potential bug today. We therefore develop a systems theory with discounting. Our theory includes several basic elements: discounted versions of system properties that correspond to the ω-regular properties, fixpoint-based algorithms for checking discounted properties, and a quantitative notion of bisimilarity for capturing the difference between two states with respect to discounted properties. We present the theory in a general form that applies to probabilistic systems as well as multicomponent systems (games), but it readily specializes to classical transition systems. We show that discounting, besides its natural practical appeal, has also several mathematical benefits. First, the resulting theory is robust, in that small perturbations of a system can cause only small changes in the properties of the system. Second, the theory is computational, in that the values of discounted properties, as well as the discounted bisimilarity distance between states, can be computed to any desired degree of precision.

It's All About Connections: Evidence on Network Formation

Abstract: We present an economic experiment on network formation, in which subjects can decide to form links to one another. Direct links are costly but being connected is valuable. The game-theoretic basis for our experiment is the model of Bala and Goyal (2000). They distinguish between two scenarios regarding the flow of benefits through a network, the so-called 1-way and 2-way flow model. Our main results show that the prediction based on Nash and strict Nash equilibrium works well in the 1-way flow model but largely fails in the 2-way flow model. We observe a strong learning dynamic in the 1-way flow model but less so in the 2-way flow model. Finally, costs of a direct link have a positive impact on the occurrence of (strict) Nash networks in the 1-way flow model but a negative impact in the 2-way flow model. In our discussion on possible explanations for these results we focus on strategic asymmetry and asymmetry with respect to payoffs. We find that the latter asymmetry, i.e. payoff inequity, plays an important role in the network formation process.

Simple Stochastic Parity Games

Abstract: Many verification, planning, and control problems can be modeled as games played on state-transition graphs by one or two players whose conflicting goals are to form a path in the graph. The focus here is on simple stochastic parity games, that is, two-player games with turn-based probabilistic transitions and ω-regular objectives formalized as parity (Rabin chain) winning conditions. An efficient translation from simple stochastic parity games to nonstochastic parity games is given. As many algorithms are known for solving the latter, the translation yields efficient algorithms for computing the states of a simple stochastic parity game from which a player can win with probability 1.

On Nash equilibria in stochastic games

Abstract: We study infinite stochastic games played by n-players on a finite graph with goals given by sets of infinite traces. The games are stochastic (each player simultaneously and independently chooses an action at each round, and the next state is determined by a probability distribution depending on the current state and the chosen actions), infinite (the game continues for an infinite number of rounds), nonzero sum (the players’ goals are not necessarily conflicting), and undiscounted. We show that if each player has a reachability objective, that is, if the goal for each player i is to visit some subset Ri of the states, then there exists an e-Nash equilibrium in memoryless strategies, for every e >0. However, exact Nash equilibria need not exist. We study the complexity of finding such Nash equilibria, and show that the payoff of some e-Nash equilibrium in memoryless strategies can be e-approximated in NP.

Knowledge sharing: a game people play

Abstract: Examines the dynamics of knowledge sharing using the multi‐person game‐theoretic framework. Proposes that an individual’s knowledge sharing tendency is driven by a set of contextualised concerns and interests not unlike the notion of payoff in game theory. Furthermore, the decision to share or withhold knowledge depends on that which yields a higher payoff. With this premise, submits two objectives. One is to investigate if an individual’s perceived payoff of sharing knowledge is contingent on the knowledge sharing behaviour of others. The other is to analyse the perceived payoff of knowledge sharing and determine if it can be characterised by an archetypical game in the game‐theoretic model. An empirical study was conducted among nearly 100 students in a local institute of higher education. The scope was confined to the students’ willingness to post asynchronous entries to an electronic discussion forum. Finds that the individual student’s perceived payoff of sharing knowledge was contingent on the knowledge sharing behaviour of others. Furthermore, the perceived payoff of knowledge sharing among them could be characterised by a multi‐person assurance game. In conclusion, discusses three implications for managers who aim to sustain asynchronous knowledge sharing in their organisations.

The Paradox of Spontaneous Formation of Private Legal Systems

Abstract: ion is made by determining a ‘payoff’ to each player (i.e., benefit conferred on the player) based on both what that player did, and what others interacting with that player do. The payoff is measured in “utils”, a generic scale measuring benefits of any kind conferred on a player (e.g., money, other material benefits, spiritual elation, a sense of being loved, etc.). There might be a different payoff to the player for each combination of her and others’ actions; mirroring real life, the choice a player makes affects her welfare, but so do the choices others make. To reduce complication, several abstractions will be made in the games examined in this paper. First, it is assumed there are two players—a network member and the other network members (or a network member and the network governance institution). Second, each player is limited a choice between two actions. These actions will change from game to game depending on the illustrative story of the game, but generally they will be called ‘Cooperate’ ({C}) and ‘Default’ ({D}). To clarify the concept of a game type, consider the ‘function’ of providing a venue for scorn: Statler and Waldorf, the grumpy old men sitting on the balcony in “The Muppets’ Show”, love to express derision at anything and everything. When the Muppets Show is not on and they find no targets for their venom on stage, they must verbally attack each other. The best that can possibly happen from Statler’s point of view is when he says something nasty to Waldorf, and Waldorf does not reply. Second to this “oneupsmanship” is a situation in which they exchange gibes. Much less satisfying is a 69 To make the illustrative examples more intuitive, this paper will sometimes call the more socially beneficial of the two actions “cooperating”, while the other, less virtuous action will be called “defaulting”. But this is not always the case. There does not have to be anything morally or socially better in an action called “cooperating” over an action called “defaulting”. In some games, the two options would be equivalent morally and from a welfare-maximizing perspective. For example, in the Battle of the Sexes game, described supra, in Section III.3, the “cooperating” action is going to see a baseball game, while the “defaulting” action is going to see a movie. The tags of cooperation and default are used merely to make this game comparable to other games discussed in this section, and not to denote a positive or negative connotation to either action. The Paradox of Spontaneous Formation of Private Legal Systems/Amitai Aviram 29 situation in which both Statler and Waldorf are nice to each other, and their venom fails to find an outlet. Bad as that sounds, it can get worse—Statler might act nicely to Waldorf, who in return will mock Statler with a nasty jeer; suffering an unanswered jab is even worse than having everyone play nice. Waldorf has the same preferences as Statler (reversing the roles, of course). Abstracting these preferences into a table, the payoff structure will look like this: Waldorf acts nicely Waldorf mocks Statler acts nicely 1,1 0,3 Statler mocks 3,0 2,2 Placing the payoff information in a table helps us identify the likely outcome. Let’s put ourselves in Statler’s shoes. If he expects Waldorf to act nicely, Statler is better off mocking him (he will then get 3 “utils” (southwest box) instead of one util (northwest box)). And if Statler expects Waldorf to mock him, Statler will—once again, mock Waldorf (he will get 2 utils (southeast box) rather than zero utils (northeast box)). So Statler will mock Waldorf regardless of what he expects Waldorf to do. Since Waldorf has the same preferences, he will reach the same conclusion, and the two will end up teasing and insulting each other. That’s good news—this happens to be the welfaremaximizing solution, since they get two utils each, or 4 total—a larger total than in any of the other boxes. This game is known as the “Deadlock” game, because if acting nicely were considered to be “cooperating”, the parties would be deadlocked in refusal to cooperate. The Deadlock game is among the most costly to enforce mutual cooperation—not only do the parties tend to not cooperate, but the welfare maximizing situation for them is 70 To summarize, the set of preferences for each player of the Deadlock game is: {D,C}>{D,D}>{C,C}>{C,D}. 71 For the payoff set in each box, Statler’s payoff is noted first, then Waldorf’s payoff. 72 A commonly cited real world example of this game would be arms control negotiations between two countries who do not want to disarm (i.e., would prefer that both they and their enemy be armed than both they and their enemy be unarmed). The likely result is a failure of the arms control negotiations. See, e.g., Janet Chen, Su-I Lu & Dan Vekhter, Game Theory—Non Zero Sum Games—Other Games, available at: http://cse.stanford.edu/classes/sophomore-college/projects-98/game-theory/dilemma.html. The Paradox of Spontaneous Formation of Private Legal Systems/Amitai Aviram 30 mutual default, so if they could coordinate, they’d attempt to enforce mutual default rather than mutual cooperation. Imagine, for example, Statler & Waldorf’s response if Kermit tried to force them to act kindly to each other... The following subsections will examine other games, their illustrative stories, their payoff structure, the likely behavior of the players and the relative ease of enforcing cooperation in them. A. Harmony The Harmony game can be seen to be an inverse of the Deadlock game. It is the easiest game in which to enforce mutual cooperation. In fact, no enforcement at all is necessary. Alice and Bill, two very good friends, face a choice between the same two actions that Statler and Waldorf chose from in the Deadlock game: they can act nicely to the other or they could mock him/her. Unlike Statler and Waldorf, each of them prefers to be nice to the other, even if he himself is slighted by the other (after all, the other’s slight may have been merely a misperception, and at any rate, they care for each other so much that hurting the other would indirectly hurt them). Next worst possibility is that they themselves somehow failed and mocked the other. In that case, each hopes that the other would show restraint and not mock back (this would be worse than being mocked while acting nicely, since the shame of being rude to one’s friend in the former case outweighs the anger at being mocked in the latter case). The worst for these two would be slipping into mutual taunting. Putting these preferences into a payoff table yields this: Bill acts nicely Bill mocks Alice acts nicely 3,3 2,1

Correlated equilibria in graphical games

Abstract: We examine correlated equilibria in the recently introduced formalism of graphical games, a succinct representation for multiplayer games. We establish a natural and powerful relationship between the graphical structure of a multiplayer game and a certain Markov network representing distributions over joint actions. Our first main result establishes that this Markov network succinctly represents all correlated equilibria of the graphical game up to expected payoff equivalence. Our second main result provides a general algorithm for computing correlated equilibria in a graphical game based on its associated Markov network. For a special class of graphical games that includes trees, this algorithm runs in time polynomial in the graphical game representation (which is polynomial in the number of players and exponential in the graph degree).

Convergent multiple-timescales reinforcement learning algorithms in normal form games

Abstract: We consider reinforcement learning algorithms in normal form games. Using two-timescales stochastic approximation, we introduce a model-free algorithm which is asymptotically equivalent to the smooth fictitious play algorithm, in that both result in asymptotic pseudotrajectories to the flow defined by the smooth best response dynamics. Both of these algorithms are shown to converge almost surely to Nash distribution in two-player zero-sum games and N -player partnership games. However, there are simple games for which these, and most other adaptive processes, fail to converge--in particular, we consider the N -player matching pennies game and Shapley's variant of the rock--scissors--paper game. By extending stochastic approximation results to multiple timescales we can allow each player to learn at a different rate. We show that this extension will converge for two-player zero-sum games and two-player partnership games, as well as for the two special cases we consider.

Isotone Equilibrium in Games of Incomplete Information

Abstract: An isotone pure strategy equilibrium exists in any game of incomplete information in which each player's action set is a finite sublattice of multidimensional Euclidean space, types are multidimensional and atomless, and each player's interim expected payoff function satisfies two “nonprimitive conditions” whenever others adopt isotone pure strategies: (i) single-crossing in own action and type and (ii) quasi-supermodularity in own action. Conditions (i), (ii) are satisfied in supermodular and log-supermodular games given affiliated types,and in games with independent types in which each player's ex post payoff satisfies supermodularity in own action and nondecreasing differences in own action and type. This result is applied to provide the first proof of pure strategy equilibrium existence in the uniform price auction when bidders have multi-unit demand, nonprivate values, and independent types.

Equilibria for Discounted Stochastic Games

Abstract: We prove the existence of subgame-perfect equilibria for discounted stochastic games with general state and action sets, using minimal assumptions (measurability as a function of states, and for each fixed state, compactness of action sets and continuity on those)—except for the rather strong assumption that the transition probabilities are norm-continuous functions of the actions.

Statistical Arbitrage and Securities Prices

Abstract: This article introduces the concept of a statistical arbitrage opportunity (SAO). In a finite-horizon economy, a SAO is a zero-cost trading strategy for which (i) the expected payoff is positive, and (ii) the conditional expected payoff in each final state of the economy is nonnegative. Unlike a pure arbitrage opportunity, a SAO can have negative payoffs provided that the average payoff in each final state is nonnegative. If the pricing kernel in the economy is path independent, then no SAOs can exist. Furthermore, ruling out SAOs imposes a novel martingale-type restriction on the dynamics of securities prices. The important properties of the restriction are that it (1) is model-free, in the sense that it requires no parametric assumptions about the true equilibrium model, (2) can be tested in samples affected by selection biases, such as the peso problem, and (3) continues to hold when investors’ beliefs are mistaken. The article argues that one can use the new restriction to empirically resolve the joint hypothesis problem present in the traditional tests of the efficient market hypothesis. In a fairly general environment, this article proposes a novel martingaletype restriction on the dynamics of securities prices. This restriction has a number of important properties. Most notably, the restriction may be viewed as model-free because it requires no parametric assumptions about the true equilibrium model. To derive the restriction, we rely on the concept of statistical arbitrage, a generalization of pure arbitrage. A pure arbitrage opportunity (PAO) is a zero-cost trading strategy that offers the possibility of a gain with no possibility of a loss. As is well known, the existence of PAOs is incompatible with a competitive equilibrium in asset markets. The fundamental theorem of the financial theory establishes a link between the absence of PAOs and the existence of a positive pricing kernel which supports securities prices. While the absence of PAOs is a necessary condition for any equilibrium model, this condition alone often yields pricing implications that are too weak to be practically useful. For example, when valuing options in incomplete markets, the no-arbitrage bounds on option prices are typically very wide. To strengthen pricing implications, several recent articles

Dynamic Equilibrium with Liquidity Constraints

Abstract: This article studies an intertemporal economy with liquidity constrained and unconstrained individuals. We use a stopping time approach to solve the finite horizonconstrained consumption portfolio problem with constant relative risk aversion and to examine the structure of equilibrium. The impact of the constraint on the optimal consumption and the financing portfolio is assessed. The equilibrium state price density is related to the exercise boundary of an American-style contingent claim with nonlinear payoff. This stopping time characterization enables us to prove the existence of an equilibrium and can be implemented numerically. Properties of equilibrium bond and stock returns are examined. Copyright 2003, Oxford University Press.

Slotted Aloha as a stochastic game with partial information

Abstract: A technique for communicating binary data comprising adding the transmitted data signals to develop a check sum, transmitting the check sum immediately following the "end-of-transmission" signal, receiving the transmitted data signals together with said check sum, adding the data signals received to develop a received signal sum, and comparing said received signal sum with said check sum to verify message integrity. Both method and apparatus are disclosed.

Phase-transition-like behaviour of quantum games

Abstract: The discontinuous dependence of the properties of a quantum game on its entanglement has been shown to be very much like phase transitions viewed in the entanglement-payoff diagram (J Du et al 2002 Phys. Rev. Lett. 88 137902). In this paper we investigate such phase-transition-like behaviour of quantum games, by suggesting a method which would help to illuminate the origin of such a kind of behaviour. For the particular case of the generalized Prisoners' Dilemma, we find that, for different settings of the numerical values in the payoff table, even though the classical game behaves the same, the quantum game exhibits different and interesting phase-transition-like behaviour.

Multi-Issue Negotiation Processes by Evolutionary Simulation, Validationand Social Extensions

Abstract: We describe a system for bilateral negotiations in which artificial agents are generated by an evolutionary algorithm (EA). The negotiations are governed by a finite-horizon version of the alternating-offers protocol. Several issues are negotiated simulataneously. We first analyse and validate the outcomes of the evolutionary system, using the game-theoretic subgame-perfect equilibrium as a benchmark. We then present two extensions of the negotiation model. In the first extension agents take into account the fairness of the obtained payoff. We find that when the fairness norm is consistently applied during the negotiation, agents reach symmetric outcomes which are robust and rather insensitive to the actual fairness settings. In the second extension we model a competitive market situation where agents have multiple bargaining opportunities before reaching the final agreement. Symmetric outcomes are now also obtained, even when the number of bargaining opportunities is small. We furthermore study the influence of search or negotiation costs in this game.