Refinement Of Nash Equilibrium Research Articles

Divide and conquer in two‐sided markets: A potential‐game approach

AbstractNetwork effects typically generate multiple equilibria in two‐sided markets. To overcome the methodological challenge of selecting an appropriate equilibrium, this article shows that many two‐sided market models are weighted potential games, and therefore, a refinement of Nash equilibrium justified by many theoretical and experimental studies, potential maximization, pins down an equilibrium. Under potential maximization, platforms often subsidize one side and charge the other, that is, divide and conquer. The primary determinant of which side to subsidize/monetize is cross‐side network effects. This divide‐and‐conquer strategy implies that platforms are often designed to favor the money side much more than the subsidy side.

The complexity of computing a (quasi-)perfect equilibrium for an n-player extensive form game

We study the complexity of computing or approximating refinements of Nash equilibrium for finite n-player extensive form games of perfect recall (EFGPR), n≥3. Our results apply to a number of well-studied refinements, including sequential equilibrium, extensive-form perfect equilibrium, and quasi-perfect equilibrium.Informally, we show that, for all these refinements, approximating such a refined equilibrium for an n-player EFGPR is not any harder than (i.e., can be efficiently reduced to) approximating a Nash equilibrium for a 3-player normal form game. More specifically, we show that approximating such a refined equilibrium for a given EFGPR, within given desired precision, is FIXPa-complete. We also study corresponding notions of “almost” equilibrium for these refinements, and we show that computing one is PPAD-complete. (In all these cases our main results show containment in FIXPa and containment in PPAD. Hardness follows from earlier results for simpler games.)

Divide and Conquer in Two-Sided Markets: A Potential-Game Approach

Network effects typically generate multiple equilibria in two-sided markets. To overcome the methodological challenge of selecting an appropriate equilibrium, this paper shows that many two-sided market models are weighted potential games, and therefore a refinement of Nash equilibrium justified by many theoretical and experimental studies, potential maximization, pins down an equilibrium. Under potential maximization, platforms often subsidize one side and charge the other, i.e., divide and conquer. The primary determinant of which side to subsidize/monetize is cross-side network effects. This divide-and-conquer strategy implies that platforms are often designed to favor the money side much more than the subsidy side.

Evolutionary stability in the generalized second-price auction

The “generalized second-price auction” is widely employed to sell internet advertising positions and has many equilibria. Analysis of this auction has assumed that myopic players commonly know each others’ position values, and that the resulting equilibrium play is “locally envy-free”. Here, I argue that the appropriate refinement of Nash equilibrium for this setting is evolutionary stability, and show that it implies that an equilibrium is locally envy-free if the whole population of players bids in each auction and the set of possible bids is not too coarse. However, not all locally envy-free equilibria are evolutionarily stable in this case, as I show by example for the popular Vickrey–Clarke–Groves outcome. The existence of evolutionarily stable equilibrium is established when one position is auctioned, as well as for two positions and a large number of bidders.

A differentiable homotopy method to compute perfect equilibria

The notion of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975) as a strict refinement of Nash equilibrium. For an extensive-form game with perfect recall, every perfect equilibrium of its agent normal-form game yields a perfect equilibrium of the extensive-form game. This paper aims to develop a differentiable homotopy method for computing perfect equilibria of normal-form games. To accomplish this objective, we constitute an artificial game by introducing a continuously differentiable function of an extra variable. The artificial game defines a differentiable homotopy mapping and establishes the existence of a smooth path to a perfect equilibrium. For numerical comparison, we also describe a simplicial homotopy method. Numerical results show that the differentiable homotopy method is numerically stable and efficient and significantly outperforms the simplicial homotopy method especially when the problem is large.

A Reformulation-Based Simplicial Homotopy Method for Approximating Perfect Equilibria

As a strict refinement of Nash equilibrium, the concept of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975). A well-known application of this concept is that every perfect equilibrium of the agent normal form game of an extensive form game with perfect recall yields a trembling-hand perfect equilibrium (consequently a sequential equilibrium). To compute a perfect equilibrium, this paper extends Kohlberg and Mertens’s equivalent reformulation of Nash equilibrium to a perturbed game. This extension naturally leads to a homotopy mapping on the Euclidean space. With this homotopy mapping and a triangulation, we develop a simplicial homotopy method for approximating perfect equilibria. It is proved that every limit point of the simplicial path yields a perfect equilibrium. Numerical results further confirm the effectiveness of the method.

Distributed Power Allocation for Wireless Sensor Network Localization: A Potential Game Approach

The problem of distributed power allocation in wireless sensor network (WSN) localization systems is investigated in this paper, using the game theoretic approach. Existing research focuses on the minimization of the localization errors of individual agent nodes over all anchor nodes subject to power budgets. When the service area and the distribution of target nodes are considered, finding the optimal trade-off between localization accuracy and power consumption is a new critical task. To cope with this issue, we propose a power allocation game where each anchor node minimizes the square position error bound (SPEB) of the service area penalized by its individual power. Meanwhile, it is proven that the power allocation game is an exact potential game which has one pure Nash equilibrium (NE) at least. In addition, we also prove the existence of an -equilibrium point, which is a refinement of NE and the better response dynamic approach can reach the end solution. Analytical and simulation results demonstrate that: (i) when prior distribution information is available, the proposed strategies have better localization accuracy than the uniform strategies; (ii) when prior distribution information is unknown, the performance of the proposed strategies outperforms power management strategies based on the second-order cone program (SOCP) for particular agent nodes after obtaining the estimated distribution of agent nodes. In addition, proposed strategies also provide an instructional trade-off between power consumption and localization accuracy.

An extension of quantal response equilibrium and determination of perfect equilibrium

As a strict refinement of Nash equilibrium, the concept of perfect equilibrium was formulated and extensively studied in the literature. To determine perfect equilibrium, this paper extends the logistic version of quantal response equilibrium (logit QRE) to a perturbed game. As a result of this extension, a smooth path is constructed for determining perfect equilibrium. The path starts from an arbitrary totally mixed strategy profile and leads to a perfect equilibrium. Numerical examples show that the extended QRE is comparable with the logit QRE and further confirm the effectiveness of the path.

Equilibrium Selection in Auctions and High Stakes Games

We introduce the test†set equilibrium refinement of Nash equilibrium to formalize the idea that players contemplate only deviations from equilibrium play in which a single competitor plays a non†equilibrium best response. We then apply this refinement to three well†known auction games, comparing our findings to similar ones previously obtained by specialized equilibrium selections. We also introduce a theory of high stakes versions of games, in which strategies are first proposed and then subjected to a potentially costly review†and†revise process. We demonstrate a sense in which the test†set equilibria emerge from such processes when the cost of revision is small.

A Rational Way of Playing: Revision Theory for Strategic Interaction

Gupta (2011) has proposed a definition of strategic rationality cast in the framework of his revision theory of truth. His analysis, relative to a class of normal form games in which all players have a strict best reply to all other players’ strategy profiles, shows that game-theoretic concepts (e.g. Nash equilibrium) have revision-theoretic counterparts. We extend Gupta’s approach to deal with normal form games in which players’ may have weak best replies. We do so by adapting intuitions relative to Nash equilibrium refinements (in particular, trembling-hand perfection and properness) to the revision-theoretic framework. We prove that there is a precise equivalence between trembling-hand perfect equilibria in two-player normal games and a revision-theoretic property. We then introduce lexicographic choice of action as a way to represent players’ expectations, which allows our analysis to reach full generality. Finally, we provide an example of the versatility of revision theory as applied to strategic interaction by formalizing a risk-and-compensation procedure of strategic choice in the revision-theoretic framework.

Irrationality and ambiguity in extensive games

Based on a model where deviations from equilibrium play are assumed to identify irrational players who are characterized by ambiguous set-valued strategies, this paper introduces an equilibrium notion for extensive games with ambiguity averse players that yields a precise interpretation for the counterfactual reasoning usually associated with backward induction. The resulting equilibria are always Nash equilibria, but may not satisfy the conditions required for various refinements of Nash equilibrium, including those for subgame perfection. Existence of such equilibria is proved for all two-player games, and for N-player games with perfect information.

Refining Nash Equilibrium by Schelling Point

This article proposes using Schelling point as a refinement of Nash equilibrium. It also introduces an algorithm way to locate Schelling points in games. The paper shows that in pure coordination games and games with significant proportion of coordination (and certain proportion of conflict), the algorithm proposed could locate the Schelling points. The existence of a Schelling point means that the proportion of common interests versus conflict of interests has crossed a certain threshold and therefore a certain form of cooperation or coexistence is possible. Besides, using Schelling point as a refinement of Nash equilibrium narrows down the set of equilibriums to only stable equilibriums as unstable equilibriums could not be Schelling points. Finally, the paper shows that the proposed approach could solve a larger set of games than the current approach based on Nash equilibrium. For instance, it would be very difficult to solve multiple sided incomplete information games under the current approach based on Nash equilibrium while the proposed approach could readily solve multiple sided incomplete information games.

Computing Nash Equilibria and Evolutionarily Stable States of Evolutionary Games

Stability analysis is an important research direction in evolutionary game theory. Evolutionarily stable states have a close relationship with Nash equilibria of repeated games, which are characterized by the folk theorem. When applying the folk theorem, one needs to compute the minimax profile of the game in order to find Nash equilibria. Computing the minimax profile is an NP-hard problem. In this paper, we investigate a new methodology to compute evolutionary stable states based on the level- ${k}$ equilibrium, a new refinement of Nash equilibrium in repeated games. A level- ${k}$ equilibrium is implemented by a group of players who adopt reactive strategies and who have no incentive to deviate from their strategies simultaneously. Computing the level- ${k}$ equilibria is tractable because the minimax payoffs and strategies are not needed. As an application, this paper develops a tractable algorithm to compute the evolutionarily stable states and the Pareto front of ${n}$ -player symmetric games. Three games, including the iterated prisoner’s dilemma, are analyzed by means of the proposed methodology.

Setting the right incentives for global planning and operations

We study incentive issues seen in a firm performing global planning and manufacturing, and local demand management. The stochastic demands in local markets are best observed by the regional business units, and the firm relies on the business units’ forecasts for planning of global manufacturing operations. We propose a class of performance evaluation schemes that induce the business units to reveal their private demand information truthfully by turning the business units’ demand revelation game into a potential game with truth telling being a potential maximizer, an appealing refinement of Nash equilibrium. Moreover, these cooperative performance evaluation schemes satisfy several essential fairness notions. After analyzing the characteristics of several performance evaluation schemes in this class, we extend our analysis to include the impact of effort on demand.

Strategic Behavior in Non-Atomic Games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player's belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.

Time and Nash implementation

In this paper, we study the full implementation problem using mechanisms that allow a delay. The delay on the equilibrium path may be zero, an infinitesimally small number or a fixed positive number. In all these three cases, implementable rules are fully characterized by a monotonicity condition. We provide examples to show that some delayed implementable social choice rules are not implementable in Nash-equilibrium refinements without a delay. As an application of our approach, we characterize delayed implementable rules in environments where only the discounting changes between states.

Direct implementation with minimally honest individuals

I consider a standard implementation problem under complete information when agents have a minimal degree of honesty. In particular, I assume that agents are white lie averse: they strictly prefer to tell the truth whenever lying has no effect on their material payoff. I show that if there are at least five agents who are all white lie averse and if I impose either of two refinements of Nash equilibrium, then a simple direct mechanism fully implements any social choice function.

Direct Implementation with Minimally Honest Individuals

I consider a standard implementation problem under complete information when agents have a minimal degree of honesty. In particular, I assume that agents are white lie averse: they strictly prefer to tell the truth whenever lying has no effect on their material payoff. I show that if there are at least five agents who are all white lie averse and if I impose either of two refinements of Nash equilibrium, then a simple direct mechanism fully implements any social choice function.

On Perfect Nash Equilibria of Polymatrix Games

When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, the perfectness concept is probably the most famous one. It is known that weakly dominated strategies of two-player games cannot be part of a perfect equilibrium. In general, this undominance property however does not extend to n-player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class of n-player games, verify the undominance property. Consequently, we prove that every perfect equilibrium of a polymatrix game is undominated and that every undominated equilibrium of a polymatrix game is perfect. This result is used to set a new characterization of perfect Nash equilibria for polymatrix games. We also prove that the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. In addition, we introduce a linear programming formulation to identify perfect equilibria for polymatrix games. These results are illustrated on two small game applications. Computational experiments on randomly generated polymatrix games with different size and density are provided.