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 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 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 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 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 . 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 ∈ SA. Action sA has a payoff dis- tributed according to a uniform distribution between 0 and i ∈ 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 , 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 . 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 ∈ 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 . 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 ∀sA, sB, sB, θA : uA((sA, sB), θA) = uA((sA, sB), θA). maxs∈S uB(s, θB) maxs∈S 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) maxs∈S uA(s, θA) A’s payoff. Obviously, player B must act according to what Proof. Let ui(θi) = maxs∈S ui(s, θi), i ∈ {A, B}. Indepen- player A chooses to do and we use sB(sA, θB) to denote the dence of player A implies that, for all θ ∈ Θ, her payoff is 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., max{uA(θA), uB(θB)} sB(sA, θB) = arg-max uB((sA, sB), θB) N ≤ P oA(θ) uA(θA) + uB(s (θ), θB) sB ∈SB 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 sA∈SA maxs∈S uB(s, θB) ≥ maxs∈S 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) ∈ arg-max EθA uB(sB(sA (θA), θB), θB) . ity by analyzing the social welfare when side payments are sB ∈SB 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. The key difference however is that, in one-way global welfare function games, the leader does not depend on the action taken by the SW ((s , s ), θ) = u (s , θ ) + u ((s , s ), θ ). follower. In addition, in one-way games, players do not have A B A A A B A B B complete information and moves are simultaneous. Jackson We quantify the quality of the Nash equilibrium outcome with and Wilkie (2005) studied one-way instances derived from the price of anarchy (PoA). their more general framework of endogenous games. How- Definition 1. The price of anarchy of sN (θ) ⊆ S given type ever, they tackled the problem from a different perspective θ is defined as and assumed complete information (i.e., the player utilities are not private). Jackson and Wilkie gave a characterization maxs∈S SW (s, θ) of the outcome when players make binding offers of side pay- P oA(θ) = . ments, deriving the conditions under which a new outcome mins∈sN (θ) SW (s, θ) becomes a Nash equilibrium or remains one. They analyzed a Throughout this paper, we use the following simple running subclass, called ’one sided externality’, which is essentially a example to illustrate key concepts. one-way game but with complete information. They showed

441 that the efficient outcome is an equilibrium in this setting, 3.1 Impossibility Result supporting Coase’s claim that a polluter and his victim can This section shows that there exists no mechanism for one- reach an efficient outcome. Under , the way games that is efficient and satisfies the traditional de- victim can determine the minimal transfer necessary to sup- sirable properties. The result is derived from the Myerson- port the efficient play. One of the key insights of this paper Satterthwaite (1983) theorem, a seminal impossibility result is the recognition that this result does not hold under incom- in mechanism design. The Myerson-Satterthwaite theorem plete information. To remedy this negative result, we present considers a bargaining game with two-sided private informa- a bargaining mechanism. tion and it states that, for a bilateral trade setting, there ex- ists no Bayes-Nash incentive-compatible mechanism that is 3 Bayesian-Nash Mechanisms budget balanced, ex-post efficient, and gives every agent type In this section, we consider a Bayesian-Nash setting with non-negative expected gains from participation (i.e., ex in- quasi-linear preferences. Both players A and B have private terim individual rationality). utilities and beliefs about the utilities of the other players. By Our contribution is twofold: we present an impossibility the revelation principle, we can restrict our attention to di- result for one-way games and we relate them with bargaining rect mechanisms which implement a social choice function. games, an idea that we will further explore on the following A social choice function in quasi-linear environments takes sections. We now formalize the impossibility result for one- the form of f(θ) = (k(θ), t(θ)) where, for every θ ∈ Θ, way games. k(θ) ∈ S is the allocation function and ti(θ) ∈ R repre- Consider the Myerson-Satterthwaite bilateral bargaining sents a monetary transfer to agent i. The main objective of setting. mechanism design is to implement a social choice function that achieves near efficient allocations while respecting some Myerson-Satterthwaite bargaining game: desirable properties. For completeness, we specify these key 1. A seller (player 1) owns an object for which properties. her valuation is v1 ∈ V1, and a buyer (player 2) wants to buy the object at a valuation v ∈ V . Definition 2. A social choice function is ex-post efficient if, 2 2 i v for all θ ∈ Θ, we have k(θ) ∈ arg-max P u (s, θ). 2. Each player knows her valuation i at the time s∈S i i of the bargaining and player 1 (resp. 2) has Definition 3. A social choice function is budget-balanced f (v ) P a probability density distribution 2 2 (resp. (BB) if, for all θ ∈ Θ, we have i ti(θ) = 0. f1(v1)) for the other player’s valuation. In other words, there are no net transfers out of the system 3. Both distributions are assumed to be continu- or into the system. Taken together, ex-post efficiency and ous and positive on their domain, and the inter- budget-balance imply Pareto optimality. An essential condi- section of the domains is not empty. tion of any mechanism is to guarantee that agents report their true types. The following property captures this notion when By the revelation principle, we can restrict our attention to agents have prior beliefs on the types of other agents. incentive-compatible direct mechanisms. A direct mechanism for bargaining games is characterized by two functions: (1) a Definition 4. A social choice function is Bayes-Nash incen- probability distribution σ : V1 × V2 → [0, 1] that specifies the tive compatible (IC) if for every player i: probability that the object is transferred from the seller to the 2 buyer and (2) a monetary transfer scheme p : V1 × V2 → R . Eθ−i|θi [ui(k(θi, θ−i), θi) + ti(θi, θ−i)] ≥ h i In this setting, ex-post efficiency is achieved if σ(v1, v2) = 1 ˆ ˆ when v < v , and 0 otherwise. Eθ−i|θi ui(k(θi, θi), θi) + ti(θi, θ−i) 1 2 Our result consists in showing that a mechanism M0 for ˆ where θi ∈ Θi is the type of player i, θi is the type player i the Myerson-Satterthwaite setting can be constructed using a mechanism M for a one-way game in such a way that, if M is reports, and Eθ−i|θi denotes player i’s expectation over prior beliefs θ−i of the types of other agents given her own type θi. efficient, individual-rational (IR), incentive compatible (IC), 0 The most natural definition of individual-rationality (IR) is in- and budget-balanced (BB), then M is efficient, IR, IC, and BB. The Myerson-Satterthwaite impossibility theorem states terim IR, which states that every agent type has non-negative 0 expected gains from participation. that such a mechanism M cannot exist, which implies the following impossibility result for one-way games. Definition 5. A social choice function is interim individual- rational if, for all types θ ∈ Θ, it satisfies Theorem 1. There is no ex-post efficient, individually ratio- nal, incentive-compatible, and budget-balanced mechanism

Eθ−i|θi [ui(k(θ), θi) + ti(θ)] ≥ ui(θi), for one-way games. where ui(θi) is the expected utility for non-participation. Proof. For any bargaining setting, consider the following In the context of one-way games, both players have posi- transformation into a one-way game instance: tive outside options that depend only in their types. In par- 1 2 ticular, the outside options are given by the Nash equilib- SA = {sA, sA}, SB = {sB}, rium outcome under no side payments. For players A and 1 2 ∀v1 ∈ V1 : uA(sA, v1) = v1, uA(sA, v1) = 0, B, the expected utilities for non-participation are uA(θA) = N N 1 2 uA(sA (θA), θA) and uB(θB) = uB(s (θ), θB) respectively. ∀v2 ∈ V2 : uB((sA, sB), v2) = 0, uB((sA, sB), v2) = v2,

442 where player types (v1, v2) ∈ V1 × V2 are drawn from distri- Given the nature of our applications, individual rationality 1 bution f1 × f2. Two possible outcomes may occur, (sA, sB) imposes a necessary constraint. Otherwise, player A can al- 2 or (sA, sB), with social welfare v1 and v2 respectively. ways defect from participating in the mechanism and achieve Let us assume M = (k, t) is a direct mechanism for her maximal payoff independently of the type of player B. one-way games and that M is ex-post efficient, IR, IC, and Additionally, we search for Bayesian-Nash mechanisms with- BB. We now construct a mechanism M0 = (σ, p), where out subsidies, i.e., budget-balanced mechanisms. The lack of σ(v1, v2) is the probability that the object is transferred from a subsidiary in this case gives rise to a decentralized mecha- the seller to the buyer and p(v1, v2) is the payment of each nism that does not require a third agent to perform the com- player. We define M0 such that putations needed by the mechanism. However, a third party is  1 needed to ensure compliance with the agreement reached by 0 if k(v1, v2) = (sA, sB), both players. σ(v1, v2) = 2 1 if k(v1, v2) = (sA, sB), An interesting starting point for one-way games is the and recognition that, whenever player B has a better payoff than A, player A may let player B play her optimal strategy in ex- p(v1, v2) = t(v1, v2). 0 change for money. The resulting outcome can be viewed as It remains to show that M satisfies all the desired prop- swapping the roles of both players, i.e., player B chooses her erties. An ex-post efficient mechanism M in the one-way in- optimal strategy and A plays her best response to B’s strat- stance satisfies egy. In this case, as in Proposition 1, the worst outcome would  1 (sA, sB) if v1 ≥ v2, be k(v1, v2) = 2 (s , sB) if v1 < v2. max u (s, θ ) A 1 + s∈S A A . max u (s, θ ) Therefore, σ(v1, v2) will assign the object to the buyer iff s∈S B B v1 < v2. That is, the player with the highest valuation will always get the object, meeting the restriction of ex-post ef- This observation leads to the following lemma. ficiency. The budget-balanced constraint in M implies that 0 Lemma 1. Consider the social choice function that selects p1(v1, v2)+p2(v1, v2) = 0 for all possible valuations, so M the best strategy that maximizes the payoff of either player A is budget-balanced. or player B, i.e., the strategy The individual rationality property for M0 comes from noticing that the default strategy of player A when no pay- 0 1 s (θ) = arg-max (max (uA(s, θA), uB(s, θB))) . ments are allowed is sA and the corresponding payoff is v1. s∈S Therefore, the seller utility is guaranteed to be at least her 0 valuation v1. Analogously, the buyer will not have a negative In the one-way game, strategy s (θ) has a price of anarchy of 1 utility given that uB((sA, sB), v2) = 0. 2 (i.e., ∀θ P oA(θ) = 2). Incentive-compatibility is straightforward from definition. Assume that M0 is not incentive-compatible, then in mech- Unfortunately, this social choice function cannot be imple- anism M, at least one player could benefit from reporting a mented in dominant strategies without violating individual false type. rationality. Player A may have a smaller payoff by follow- Such a mechanism M0 cannot exist since it contradicts ing strategy s0 instead of the Nash equilibrium strategy sN . Myerson-Satterthwaite impossibility result, which concludes Indeed, when SW (s0, θ) < SW (sN , θ), it must be that at our proof. least one of the players will be worse than playing the Nash equilibrium strategy sN . Lemma 1 however gives us hope for An immediate consequence of this result is that Bayesian- designing a budget-balanced mechanism that has a constant Nash mechanisms can only achieve at most two of the three price of anarchy. Indeed, a simple and distributed implemen- properties: ex-post efficiency, individual-rationality, and bud- tation would ask player B to propose an action to be imple- get balance. For instance, VCG and dAGVA [d’Aspremont mented and player A would receive a monetary compensation and Gerard-Varet,´ 1979; Arrow, 1979] are part of the Groves for deviating from her maximal strategy. family of mechanisms that truthfully implement social choice We now present such a distributed implementation based functions that are ex-post efficient. VCG has no guarantee on a bargaining mechanism. The mechanism is inspired by of budget balance, while dAGVA is not guaranteed to meet the model of two-person bargaining under incomplete infor- the individual-rationality constraints. We refer the reader to mation presented by Chatterjee and Samuelson (1983). In Williams (1999) and Krishna and Perry (1998) for alterna- their model, both the seller and the buyer submit sealed of- tive derivations of the impossibility result for bilateral trading fers and a trade occurs if there is a gap in the bids. The price under the Groves family of mechanisms. is then set to be a convex combination of the bids. Our single- offer mechanism adapts this idea to one-way games. In par- 4 Single-Offer Mechanism ticular, to counteract player A’s advantage, player B makes In this section, we propose a simple bargaining mechanism the first and final offer. Moreover, the structure of our mech- for player B to increase her payoff. The literature about bar- anism makes it possible to quantify the price of anarchy and gaining games is extensive and we refer readers to a broad provide quality guarantee on the mechanism outcome. Our review by Kennan and Wilson (1993). single-offer mechanism is defined as follows:

443 Single-offer mechanism: an agreement, player B is left with a profit of 1. Player B selects an action sA ∈ SA to propose u (s (s , θ ), θ ) − γ · ∆ (s , θ ). to player A. B B A B B B A B N 2. Player B also computes her expected outside Otherwise, player B gets an expected payoff of uB (θB). option uN (θ ) = u (sN (θ), θ ). B B EθA|θB B B Definition 6. The expected profit of players A and B for pro- 3. Player B proposes a monetary value of posed action s = (sA, sB) and γ when player B has type θB γ · ∆B(sA, θB) with ∆B(sA, θB) = N is given by uB(sB(sA, θB), θB) − uB (θB) and γ ∈ R[0,1] to player A in the hope that she accepts to play N θA [UB(sA, γ, θB)] = uB (θB)+ N E strategy sA instead of strategy sA . P (s , γ, θ ) ((1 − γ) · ∆ (s , θ )) , 4. Player A decides whether to accept the offer. A B B A B N 5. If player A accepts the offer, the outcome of EθA [UA(sA, γ, θB)] = EθA [uA (θA)] + P (sA, γ, θB)· the game is (s , s (s , θ )); Otherwise the A B A B (γ∆B(sA, θB) − EθA [∆A(sA, θA)]) , outcome of the game is the Nash equilibrium N N  where the probability that player A accepts to play s is de- sA (θA), sB (θB) . A fined by It is worth observing that a broker is required in this mecha- N N  P (sA, γ, θB) = Pr [γ · ∆B(sA, θB) ≥ ∆A(sA, θA)] . nism to ensure that the outcome sA (θA), sB (θB) is imple- mented if player A rejects the unique offer, and no counterof- The optimal strategy of player B is specified in the following fers are made. A key feature of the single-offer mechanism is lemma. that it requires a minimum amount of information from player A (i.e., whether she accepts or rejects the offer). Lemma 2. On the single-offer mechanism, player B chooses ∗ ∗ ∗ s (θB) and γ (s , θB) such that Proposition 2. If players A and B play the single-offer mech- A A anism, player A accepts the offer whenever ∗ sA(θB) = arg-max s N A uA(sA, θA) + γ · ∆B(sA, θB) ≥ uA(sA (θA), θA). ∗ ∗ P (sA, γ (sA, θB), θB) · (1 − γ (sA, θB)) · ∆B(sA) We have designed a mechanism that satisfies individual ra- where tionality: Player B never offers more than ∆B(sA, θB) and ∗ her payoff is never worse than her expected outside option. γ (sA, θB) = arg-max P (sA, γ, θB) · (1 − γ). By Proposition 2, player A is always better off playing the γ single-offer mechanism. 4.1 Price of Anarchy Example 2. (Example 1 continued) The payoff of Player B 1 We now analyze the quality of the outcomes in the single- is higher if action sA is played by player A. Hence, player 1 offer mechanism. B has incentives to submit an offer c that triggers action sA. 1 2 The first step is the derivation of a lower bound for the Player A accepts the offer if c + uA(sA) ≥ uA(sA) = 100. 1 expected social welfare of the single offer mechanism. In- Given that uA(sA) follows a uniform distribution, the proba- c spired by Lemma 1, instead of considering all pairs hsA, γi, bility that player A accepts the offer is 100 and such an offer 0 c the analysis restricts attention to a single action sA = has an expected payoff of 100 · (x − c) for player B. The op- x ∗ arg-maxsA∈SA uB(sB(sA, θB), θB). We prove that, when timal value for the offer is given by c = 2 . This leads to an 0 c∗ c∗ offering to player A action sA and its associated optimal value expected social welfare SW = 100 ·(50+x)+(1− 100 )·100 for γ, the expected social welfare is lower than the optimal for the single-offer mechanism. Recall that the optimal social ∗ ∗ pair hsA, γ i. As a result, we obtain an upper bound to the welfare is 50 + x if x ≥ 50 and 100 otherwise. Therefore, price of anarchy of the single-offer mechanism. 50+x ≤ 1.21 the mechanism has a price of anarchy of SW if To make the discussion precise, consider the strategy where x ≥ 50 and 100 ≤ 1.04 otherwise. This contrasts with the 0 ∗ 0 ∗ 0 SW player B offers hsA, γ (sA, θB)i, with γ (sA, θB) being the unbounded PoA obtained by the Nash equilibrium when no 0 optimal choice of γ given sA, following the notation used in side payments are allowed. Lemma 2. We now generalize the analysis done in Example 2. We Lemma 3. The expected social welfare achieved by the sin- proceed by studying the utility-maximizing strategy (sA, γ) gle offer mechanism is at least the expected social welfare for player B and then derive the expected social welfare of the 0 ∗ 0 achieved by the strategy hsA, γ (sA, θB)i. outcome for the single-offer mechanism. Note that, in case ∗ ∗ ∗ 0 ∗ 0 of agreement, the action of player B of type θB is solely Proof. Let γ = γ (sA, θB) and γ = γ (sA, θB). The opti- ∗ defined by sA as she has no incentives to defect from its mality condition of s implies that best response sB(sA, θB). By Proposition 2, player A ac- 0 0 ∗ ∗ θ [UB(s , γ , θB)] ≤ θ [UB(s , γ , θB)] . (1) cepts an offer whenever ∆A(sA, θA) ≤ γ∆B(sA, θB), where E A E A N ∆A(sA, θA) = uA(sA (θA), θA) − uA(sA, θA). Player B ob- Two cases can occur. The first case is viously aims at choosing γ and sA to maximize her payoff 0 0 ∗ ∗ and we now study this optimization problem. In the case of P (sA, γ , θB) ≤ P (sA, γ , θB),

444 ∗ ∗ 0 0 N 0 0 i.e., the probabilty of player A accepting offer (sA, γ ) is Case uA + γ∆B(sA) ≥ uA . Strategy (sA, sB) is played. 0 0 greater than if offered (sA, γ ). Then, it must be that the ex- ∗ ∗ uN + u0 u0 + u0 + γ · u0 pected payoff of player A is greater when offered (sA, γ ), A A B A B B P oA ≤ 0 0 ≤ 0 0 i.e., uA + uB uA + uB 0 0 0 ∗ ∗ uB θ [UA(s , γ , θB)] ≤ θ [UA(s , γ , θB)] . E A A E A A = 1 + γ 0 0 ≤ 1 + γ. uA + uB This, together with Inequality (1) results in the single-offer 0 0 N N mechanism having a greater expected social welfare. Case uA + γ∆B(sA) < uA . Player A plays sA . N 0 N ∗ ∗ The second case is γ · uB ≤ uA + γ · uB < uA + γ · uB ≤ uA + uB then, 0 0 ∗ ∗ N 0 0 P (sA, γ , θB) > P (sA, γ , θB). R uA + uB uB P oA ≤ N N ≤ 1 + N N 00 0 00 ∗ ∗ uA + uB uA + uB Consider γ such that P (sA, γ , θB) = P (sA, γ , θB). The fact that the probabilities of acceptance are the same implies u0 1 ≤ 1 + B = 1 + . that the expected payoff of Player A is the same in both cases, γ · u0 γ 0 00 ∗ ∗ B i.e., EθA [UA(sA, γ , θB)] = EθA [UA(sA, γ , θB)]. This, to- gether with Equation (1) yields ∗ ∗ 0 00 When γ = 1, the price of anarchy is 2 but player B has no EθA [SW (s , γ , θB)] ≥ EθA [SW (s , γ , θB)]. incentive to choose such a value. If γ = 0.5, the price of an- This is equivalent to 0 ∗ 0 archy is 3. Of course, player B will choose γ = γ (sA, θB). Lemma 4 indicates that the worst-case outcome is (1 + γ0) ∗ ∗ uB(s , θB) + θ [uA(s , θA)] ≥ 0 0 E A A when player A accepts with a probability P (sA, γ , θB) and 0 0 (1 + 1 ) uB(s , θB) + EθA [uA(sA, θA)]. (2) γ0 otherwise. This yields the following result. Similarly, consider γ∗∗ such that Theorem 2. The Bayesian price of anarchy for one-way games is at most P (s0 , γ0, θ ) = P (s∗ , γ∗∗, θ ), A B A B γ0 + 1 (1 − P (s0 , γ0, θ )(1 − γ0)) , which implies γ0 A B 0 0 ∗ ∗∗ θA [UA(sA, γ , θB)] = θA [UA(sA, γ , θB)] . where E E 0 0 γ = arg-max P (sA, γ, θB)(1 − γ). Existence of γ∗∗ is guaranteed by Inequality (2) which states γ that, there is more money in expectation to transfer to player To get a better idea of how the mechanism improves the so- ∗ 0 A when choosing s over s . The fact that the acceptance cial welfare, it is useful to quantify the price of anarchy in probabilities are the same, together with Inequality (2), im- Theorem 2 for a specific class of distributions. plies that 0 Corollary 1. If ∆A(sA, θA) has a cumulative distribution ∗ ∗∗ 0 0 β EθA [SW (s , γ , θB)] ≥ EθA [SW (s , γ , θB)]. function F (x) = (x/∆B) between 0 and ∆B, with 0 < β ≤ 1, then γ = β and the price of anarchy is at most Given that the expected payoff of player A is the same in both β+1 cases, it must be the case that the expected payoff of player 1 ∗ ∗∗ (2 + )(1 − ββ(1 + β)−(β+1)). B is higher when using (sA, γ ). β Therefore, we have found an offer for the single-offer mechanism with greater expected social welfare and a greater For example, if β = 1, then F (x) is the uniform distribution, 0 0 1 payoff for player B compared with strategy hsA, γ i. γ = 2 , and the expected price of anarchy is at most 2.25. This corollary, in conjunction with Lemma 1, gives us the price We are ready to derive an upper bound for the induced cost of enforcing individual rationality, moving from a price of anarchy for the single-offer mechanism. We first derive the of anarchy of 2 to a price of 2.25 in the case of a uniform hs0 , γ0i price of anarchy of strategy A in case of agreement and distribution. disagreement of player A. 0 0 The strategy hsA, γ i is of independent interest. It indicates 0 Lemma 4. Consider action s = arg maxs∈S uB(s, θB) and how a player with limited computational power can achieve let P oAA(γ) and P oAR(γ) denote the induced price of anar- an outcome that satisfies individual rationality without opti- chy if player A accepts and rejects the offer given a proposed mizing over all strategies. γ. Then, 1 5 Conclusion P oAA(γ) = 1 + γ and P oAR(γ) = 1 + . γ In one-way games, the utility of one player does not depend on the decisions of the other player. We showed that, in this N N 0 0 Proof. Define uA = uA(sA , θA), uA = uA(s , θA) and setting, the outcome of a Nash equilibrium can be arbitrar- 0 0 uB = uB(s , θB). Two cases can occur. ily far from the social welfare solution. We also proved that

445 it is impossible to design a Bayes-Nash incentive-compatible [d’Aspremont and Gerard-Varet,´ 1979] Claude mechanism for one-way games that is budget-balanced, in- d’Aspremont and Louis Andre Gerard-Varet.´ Incen- dividually rational, and efficient. To alleviate these negative tives and incomplete information. Journal of Public results, we proposed a privacy-preserving mechanism based Economics, 11:25–45, 1979. on a single-offer. [Hahnel and Sheeran, 2009] Robin Hahnel and Kristen A The single-offer mechanism is simple for both parties, as Sheeran. Misinterpreting the coase theorem. Journal of well as for the broker who just makes sure that the players Economic Issues, 43:215–238, 2009. follow the protocol. This mechanism also requires minimal information from the agents who perform all the combina- [Jackson and Wilkie, 2005] Matthew O Jackson and Simon torial computations, while it incentivizes them to cooperate Wilkie. Endogenous games and mechanisms: Side pay- towards the social welfare in a distributed setting. Moreover, ments among players. The Review of Economic Studies, the mechanism has the following desirable properties: It is 72:543–566, 2005. budget-balanced and satisfies the individual rationality con- [Kennan and Wilson, 1993] John Kennan and Robert Wil- straints and Bayesian incentive-compatibility conditions. Ad- son. Bargaining with private information. Journal of Eco- ditionally, we showed that, in a realistic setting, where agents nomic Literature, 31:45–104, 1993. have limited computational resources, a simpler version of the [Krishna and Perry, 1998] Vijay Krishna and Motty Perry. mechanism can be implemented without overly deteriorating Efficient mechanism design. Available at SSRN 64934, the social welfare. 1998. It is an open question whether there exists another mecha- nism (possibly more complex) that could lead to a better ef- [Myerson and Satterthwaite, 1983] Roger B Myerson and ficiency, while keeping the above properties. Indeed, in one- Mark A Satterthwaite. Efficient mechanisms for bilateral way games, player A has a intrinsic advantage over player B, trading. Journal of Economic Theory, 29:265–281, 1983. which is not easy to overcome. There are also many other di- [Van Hentenryck et al., 2010] Pascal Van Hentenryck, Rus- rections for future research. It is important to generalize one- sell Bent, and Carleton Coffrin. Strategic planning for way games to multiple players. Moreover, there are applica- disaster recovery with stochastic last mile distribution. In tions where the dependencies are in both directions, e.g., the CPAIOR, pages 318–333. Springer, 2010. restoration of the power and the gas systems considered in [Voigt, 2011] Guido Voigt. Supply Chain Coordination in Coffrin et al. (2012). These applications typically have mul- Case of Asymmetric Information. Springer, 2011. tiple components to restore and the dependencies form an acyclic graph. Hence such a mechanism would likely need to [Williams, 1999] Steven R Williams. A characterization of consider this internal structure to obtain efficient outcomes. efficient, bayesian incentive compatible mechanisms. Eco- nomic Theory, 14(1):155–180, 1999. Acknowledgments NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.

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