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A Bargaining Mechanism for One-Way Games Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) A Bargaining Mechanism for One-Way Games Andres´ Abeliuk Gerardo Berbeglia Pascal Van Hentenryck NICTA and NICTA and NICTA and University of Melbourne Melbourne Business School Australian National University [email protected] [email protected] [email protected] Abstract information plays an important role and game theory and bar- gaining games can explain inefficient outcomes. We introduce one-way games, a framework moti- vated by applications in large-scale power restora- Other real-life applications are found in large-scale restora- tion, humanitarian logistics, and integrated supply- tion of interdependent infrastructures after significant disrup- chains. The distinguishable feature of the games tions [Cavdaroglu et al., 2013; Coffrin et al., 2012], human- is that the payoff of some player is determined itarian logistics over multiple states or regions [Van Hen- only by her own strategy and does not depend on tenryck et al., 2010], supply chain coordination (see, e.g., actions taken by other players. We show that the Voigt (2011)), integrated logistics, and the joint planning and equilibrium outcome in one-way games without the control of gas and electricity networks. Consider, for in- payments and the social cost of any ex-post effi- stance, the restoration of the power system and the telecom- cient mechanism, can be far from the optimum. We munication network after a major disaster. As explained in also show that it is impossible to design a Bayes- [Cavdaroglu et al., 2013], there are one-way dependencies Nash incentive-compatible mechanism for one-way between the power system and the telecommunication net- games that is budget-balanced, individually ratio- work. This means, for instance, that some power lines must nal, and efficient. Finally, we propose a privacy- be restored before some parts of the telecommunication net- preserving mechanism that is incentive-compatible work become available. It is possible to use centralized mech- and budget-balanced, satisfies ex-post individual anisms for restoring the system as a whole. However, it is rationality conditions, and produces an outcome often the case that these restorations are performed by differ- which is more efficient than the equilibrium with- ent agencies with independent objectives and selfish behavior out payments. may have a strong impact on the social welfare. It is thus im- portant to study whether it is possible to find high-quality out- comes in decentralized settings when stakeholders proceed 1 Introduction independently and do not share complete information about When modeling economic interactions between agents, it is their utilities. standard to adopt a general framework where payoffs of indi- This paper aims at taking a first step in this direction by viduals are dependent on the actions of all other decision- proposing a class of two players one-way dependent deci- makers. However, some agents may have payoffs that de- sion settings which abstracts some of the salient features pend only on their own actions, not on actions taken by other of these applications and formalizes many of Hahnel and agents. In this paper, we explore the consequences of such Sheeran’s critiques. We present a number of negative and asymmetries among agents. Since these features lead to a re- positive results on one-way games. We first show that Nash stricted version of the general model, the hope is that we can equilibria in one-way games, under no side payments, can identify mechanisms that produce efficient outcomes by ex- be arbitrarily far from the optimal social welfare. More- ploiting the properties of this specific setting. over, in contrast to Coase theorem, we show that when side A classic application of this setting is Coase’s example of payments are allowed in a Bayes-Nash incentive-compatible a polluter and a single victim, e.g., a steel mill that affects setting, there is no ex-post efficient individually rational, a laundry. The Coase theorem [Coase, 1960] is often inter- and budget-balanced mechanism for one-way games. To ad- preted as a demonstration of why private negotiations be- dress this negative result, we focus on mechanisms that are tween polluters and victims can yield efficient levels of pol- budget-balanced, individually rational, incentive-compatible, lution without government interference. However, in an influ- and have a small price of anarchy. Such mechanisms are par- ential article, Hahnel and Sheeran (2009) criticize the Coase ticularly useful for rational agents who have incomplete in- theorem by showing that, under more realistic conditions, it formation about other agents. Our main positive result is a is unlikely that an efficient outcome will be reached. They single-offer bargaining mechanism which under reasonable emphasize that the solution is a negotiation, and not a market- assumptions on the players, increases the social welfare com- based transaction as described by Coase. As such, incomplete pared to the setting where no side payments are allowed. 440 2 One-Way Games Example 1. Consider the instance where player A has two 1 2 1 One-way games feature two players A and B. Each player possible actions sA; sA 2 SA. Action sA has a payoff dis- tributed according to a uniform distribution between 0 and i 2 A; B has a public action set Si and we write S = SA×SB 2 to denote the set of joint action profiles. As most commonly 100, while action sA has a constant payoff of 100. The set of dominant actions for player B corresponds to the set of done in mechanism design, we model private information by 1 2 best responses, i.e., sB(sA) and sB(sA). We set payoffs to be associating each agent i with a payoff function ui : S ×Θi ! 1 2 +, where u (s; θ ) is the agent utility for strategy profile s uB(sB(sA)) = x and uB(sB(sA)) = 0, where x is a pos- R i i itive constant. When no transfers are allowed, player A will when the agent has type θi. We assume that the player types 2 are stochastically independent and drawn from a distribution always play action sA, yielding a social welfare of 100 + 0. If player A plays s1 , her expected payoff is 50 and the expected f that is common knowledge. We denote by Θi the possible A social welfare is 50 + x. Thus, the price of anarchy is 50+x types of player i and write Θ = ΘA × ΘB. If θ 2 Θ, we use 100 if x ≥ 50 and 1 otherwise. Notice that the PoA is unbounded θi to denote the type of player i in θ. Similar conventions are used for strategies, utilities, and type distributions. on x for the Nash equilibrium. A key feature of one-way games is that the payoff We now quantify the price of anarchy in one-way games. uA((sA; sB); θA) of player A is determined only by her own Proposition 1. In one-way games, the price of anarchy when strategy and does not depend on B’s actions, i.e., no payments are allowed satisfies, for any type θ, 0 0 8sA; sB; sB; θA : uA((sA; sB); θA) = uA((sA; sB); θA): maxs2S uB(s; θB) maxs2S uB(s; θB) N ≤ P oA(θ) ≤ 1 + : As a result, for ease of notation, we use uA(sA; θA) to denote SW (s (θ); θ) maxs2S uA(s; θA) A’s payoff. Obviously, player B must act according to what Proof. Let ui(θi) = maxs2S ui(s; θi), i 2 fA; Bg. Indepen- player A chooses to do and we use sB(sA; θB) to denote the dence of player A implies that, for all θ 2 Θ, her payoff is best response of player B given that A plays action s and N A uA(s (θ); θA) = uA(θA). It follows that player B has type θB, i.e., maxfuA(θA); uB(θB)g sB(sA; θB) = arg-max uB((sA; sB); θB) N ≤ P oA(θ) uA(θA) + uB(s (θ); θB) sB 2SB uA(θA) + uB(θB) uA(θA) + uB(θB) where ties are broken arbitrarily. In this paper, we always as- ≤ N ≤ sume that ties are broken arbitrarily in arg-max expressions. uA(θA) + uB(s (θ); θB) uA(θA) One-way games assumes that players simultaneously uB(θB) = 1 + : choose their actions and players are risk-neutral agents. uA(θA) Hence, if side payments are not allowed, player A will play N an action sA that yields her a maximum payoff, i.e., N The price of anarchy can thus be arbitrarily large. sA (θA) = arg-max uA(sA; θA); When it is large enough, Proposition 1 indicates that sA2SA maxs2S uB(s; θB) ≥ maxs2S uA(s; θA). In this case, player N Player B will pick sB (θB) such that her expected payoff is B has bargaining power to incentivize player A monetarily so maximized, i.e., that she moves from her equilibrium and cooperates to over- N N come a bad social welfare. This paper explores this possibil- sB (θB) 2 arg-max EθA uB(sB(sA (θA); θB); θB) : ity by analyzing the social welfare when side payments are sB 2SB allowed. The set of Nash equilibria (NE) is thus characterized by N N N N s (θ) = (sA (θA); sB (θB)) ⊆ S. The best response sB (θB) Related Work Before moving to the main results, it is of player B may be a bad outcome for her even when B has a useful to discuss related games. One-way games may seem much greater potential payoff. Player A achieves its optimal to resemble Stackelberg games with their notions of leader payoff, but our motivating applications aim at optimizing a and follower.
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