# Research

### Research

### Working Papers

**A Stochastic Stability Analysis with Observation Errors in Normal Form Games **[pdf]

We perform a stochastic stability analysis with observation errors. Players recurrently play a symmetric two-player normal form game with one another and respond to the strategy distribution of other players. In each period, a revising player observes the strategy distribution and chooses a best response to it. Her observation is perturbed with positive probability and she may respond to the misperceived strategy distribution. We examine the robustness of Nash equilibria to such observation errors. We find that the set of stochastically stable states under observation errors is robust to addition of strictly dominated strategies for a certain class of games given that observation errors are uniform, i.e., each misrepresented state is observed with uniform probability. We also examine the set of stochastically stable states under an alternate observation error model where the observation probability depends on the L1-norm between the true state and the observed state. For the local interaction model, we characterize conditions for each error model under which the set of stochastically stable states is robust to addition of strategies that do not survive the iterative elimination of strictly dominated strategies.

**The Evolution of Collective Choice under Majority Rule (with Akira Okada) **[pdf]

We consider a dynamic process of collective choice under majority rule in which a status quo policy evolves. The analysis is based on stochastic evolutionary game theory. The Condorcet winner is uniquely a long-run equilibrium for all (super-)majority rules. When the Condorcet winner does not exist, the long-run equilibria under all majority voting rules belong to the top cycle of policies under a simple majority. When the policy space is multidimensional and the voting quota is larger than the min-max quota, the long-run equilibrium belongs to the minmax set. Finally, the Borda winner appears as a long-run equilibrium under unanimity if votersâ€™ behavior is governed by a logit choice rule.

**Stochastic Stability in the Large Population and Small Mutation Limits for Coordination Games **[pdf]

We consider a model of stochastic evolution in symmetric coordination games with K>=2 strategies played by myopic agents. Agents employ the best response with mutations choice rule and simultaneously revise strategies in each period. We form the dynamic process as a Markov chain with state space being the set of best responses in order to overcome difficulties that arise with the large population. We examine the long run equilibria for both orders of limits where the small noise limit and the large population limit are taken sequentially. We characterize an equilibrium refinement criterion that is common among both orders of limits.

### Published Papers

**A prospect theory Nash bargaining solution and its stochastic stability**

*Journal of Economic Behavior and Organization**, Forthcoming. *

We consider the long-run outcomes of bargaining games when players obey prospect theory. We extend the evolutionary bargaining model of Young (1993) to a two-stage Nash demand game. Two players simultaneously choose whether to exercise an outside option in the first stage and play the Nash demand game in the second stage, which will be reached only if neither player exercises the outside option. We address the influence on the stochastically stable division of reference-dependent preferences where the reference point is the value of the outside option. We show that the division consistently differs from the Nash bargaining solution under expected utility theory. Inspired by this, we propose a prospect theory Nash bargaining solution, which coincides with the stochastically stable division.

**Evolutionary dynamics in multitasking environments (with Dai Zusai)**

*Journal of Economic Behavior and Organization 166**, pp.288--308 (2019) *

We formulate the best response dynamic in a multitasking environment; while agents engage in multiple games concurrently, an agent can switch her action in only one of the games upon receipt of a revision opportunity. The choice of the game in which to revise an action makes the multitasking dynamic behave differently from standard evolutionary dynamics. The timing of revisions in a game becomes endogenous, which causes the transition of the action distribution in each game to depend on those in other games. Despite such complexity, we verify the global stability of the Nash equilibrium set in potential and contractive games as well as the local stability of a regular evolutionary stable state. We also show that the equilibrium to which the multitasking dynamic converges may depend on the task choice rules.

**Stochastic stability under logit choice in coalitional bargaining problems [****pdf****]**

*Games and Economic Behavior 113**, *pp.633--650 (2019)

This study examines a dynamic process of n-person coalitional bargaining problems. We investigate the evolution of social conventions by embedding a coalitional bargaining setting in a dynamic process. Under a logit specification of choice probabilities, we find that the stability of a core allocation decreases in the wealth of the richest player. Furthermore, stochastically stable allocations are core allocations that minimize the wealth of the richest player.

**Prospect Dynamic and Loss Dominance (with Jiabin Wu) ** [pdf]

*Games and Economic Behavior 112**, *pp.98--124 (2018)

This paper investigates the role of loss-aversion in affecting the long-run equilibria of stochastic evolutionary dynamics. We consider a finite population of loss-averse agents who are repeatedly and randomly matched to play a symmetric two-player normal form game. We propose a stronger concept, *loss-dominance*: a strategy is loss-dominant if it is risk-dominant and a maximin strategy. In 2x2 coordination games, the state where all agents play the loss-dominant strategy is uniquely stochastically stable under prospect dynamics for any degree of loss-aversion and all types of reference points.

**Reference-dependent preferences, super-dominance and stochastic stability (with Jiabin Wu) **

*Journal of Mathematical Economics 78**, *pp.96--104 (2018)

This paper investigates stochastic stability of noisy best response dynamics with reference-dependent preferences. We define a strategy as *super-dominant* in a 2x2 coordination game if it is the maximin strategy in terms of monetary returns and the state that all players play it constitutes an equilibrium which Pareto-dominates all other equilibria. If such a strategy exists, the corresponding equilibrium, which we call the super-dominant equilibrium, is uniquely stochastically stable for the BRM choice rule (the best response choice rule with uniform random errors) given any model of reference-dependent preferences.

**A one-shot deviation principle for stability in matching problems (with Jonathan Newton) ** [pdf]

*Journal of Economic Theory 157**, *pp.1--27 (2015)

This paper considers marriage problems, roommate problems with nonempty core, and college admissions problems with responsive preferences. All stochastically stable matchings are shown to be contained in the set of matchings which are most robust to one-shot deviation.

**Evolutionary imitative dynamics with population-varying aspiration levels (with Dai Zusai)** [pdf]

*Journal of Economic Theory 154**, *pp.562--577 (2014)

We consider deterministic evolutionary dynamics under imitative revision protocols. We allow agents to have different aspiration levels in the imitative protocols where their aspiration levels are not observable to other agents. We show that the distribution of strategies becomes statistically independent of the aspiration level eventually in the long run. Thus, long-run properties of homogeneous imitative dynamics hold as well, despite heterogeneity in aspiration levels.

**Coalitional stochastic stability in games, networks and markets** [pdf]

*Games and Economic Behavior 88**, *pp.90--111 (2014)

This paper examines a dynamic process of unilateral and joint deviations of agents and the resulting stochastic evolution of social conventions. Our model unifies stochastic stability analysis in static settings, including normal form games, network formation games, and simple exchange economies, as stochastic stability analysis in a class of interactions in which agents unilaterally and jointly choose their strategies. We embed a static setting in a dynamic process; Over time agents revise their strategies based on the improvements that the new strategy profile offers them. In addition to the optimization process, there are persistent random shocks on agents utility that potentially lead to switching to suboptimal strategies. Under a logit specification of choice probabilities, we characterize the set of states that will be observed in the long-run as noise vanishes. We apply these results to examples of certain potential games.

**Mutation Rates and Equilibrium Selection under Stochastic Evolutionary Dynamics [pdf]**

*International Journal of Game Theory 41**, *pp.489--496 (2012)

Bergin and Lipman (1996) show that equilibrium selection using stochastic evolutionary processes depends on the specification of mutation rates. We offer a characterization of how mutation rates determine the selection of Nash equilibria in 2x2 symmetric coordination games for single and double limits of the small mutation rates and the large population size.

Copyright 2012 Ryoji Sawa

University of Tsukuba

Contact: rsawa[at]sk.tsukuba.ac.jp