Robust Quantitative Comparative Statics for a Multimarket Paradox

We introduce a quantitative approach to comparative statics that allows to bound the maximum effect of an exogenous parameter change on a system's equilibrium. The motivation for this approach is a well known paradox in multimarket Cournot competition, where a positive price shock on a monopoly market may actually reduce the monopolist's profit. We use our approach to quantify for the first time the worst case profit reduction for multimarket oligopolies exposed to arbitrary positive price shocks. For markets with affine price functions and firms with convex cost technologies, we show that the relative profit loss of any firm is at most 25% no matter how many firms compete in the oligopoly. We further investigate the impact of positive price shocks on total profit of all firms as well as on social welfare. We find tight bounds also for these measures showing that total profit and social welfare decreases by at most 25% and 16.6%, respectively. Finally, we show that in our model, mixed, correlated and coarse correlated equilibria are essentially unique, thus, all our bounds apply to these game solutions as well.Comment: 23 pages, 1 figur

Equilibrium Design by Coarse Correlation in Quadratic Games

In a public good provision or a public bad abatement situation, the non-cooperative interplay of the participants typically results in low levels of provision or abatement. In the familiar class of n-person quadratic games, we show that Coarse Correlated equilibria (CCEs) - simple mediated communication devices that do not alter the strategic structure of the game - can significantly outperform the Nash equilibrium in terms of the policy objective above

Correlated Equilibrium Under Costly Disobedience

In this paper, we extend Aumann's (1974) well-known solution of correlated equilibrium to allow for a cost of disobedience for each player. Calling the new solution costly correlated equilibrium (CCE), we derive the necessary and sufficient conditions under which the set of CCE strictly expands when the players' cost of disobedience is increased by the mediator in any finite normal-form game. These conditions imply that for any game that has a Nash equilibrium (NE) that is unpure, the set of CCE strictly expands with the addition of even arbitrarily small cost of disobedience, whereas for games that have a unique NE in pure strategies, the set of CCE stays the same unless the cost gets sufficiently high. We also study the welfare implications and changes in the value of mediation with exogenous cost changes. We find that strictly better social outcomes can be attained and the value of mediation cannot decrease with an increase in the cost level. We also illustrate how our model can be integrated with a cost-selection game where players non-cooperatively choose their costs of disobedience before mediation occurs. We show that there exist cost-selection games in which setting the cost of disobedience at zero is a strictly dominated strategy for each player as well as games this strategy becomes weakly dominant for everyone

Distribution-Valued Solution Concepts

Under its conventional positive interpretation, game theory predicts that the mixed strategy pro?le of players in a noncooperative game will satisfy some setvalued solution concept. Relative probabilities of pro?les in that set are unspeci?ed, and all pro?les not satisfying it are implicitly assigned probability zero. However the axioms underlying Bayesian rationality say that we should reason about player behavior using a probability density over all mixed strategy pro?les, not using a subset of all such pro?les. Such a density over pro?les can be viewed as a solution concept that is distribution-valued rather than set-valued. A distribution-valued concept provides a best single prediction for any noncooperative game, i.e., a universal re?nement. In addition, regulators can use a distribution-valued solution concept to make Bayes optimal choices of a mechanism, as required by Savage's axioms. In particular, they can do this in strategic situations where conventional mechanism design cannot provide advice. We illustrate all of this on a Cournot duopoly game.Quantal Response Equilibrium, Bayesian Statistics, Entropic prior, Maximum entropy JEL Codes: C02, C11, C70, C72

Coarse correlated equilibria in an abatement game

We consider the well-analyzed abatement game (Barrett 1994) and prove that correlation among the players (nations) can strictly improve upon the Nash equilibrium payoffs. As these games are potential games, correlated equilibrium — CE — (Aumann 1974, 1987) cannot improve upon Nash; however we prove that coarse correlated equilibria — CCE — (Moulin and Vial 1978) may do so. We compute the largest feasible total utility and hence the efficiency gain in any CCE in those games: it is achieved by a lottery over only two pure strategy profiles

Coarse correlation and coordination in a game:an experiment

In a coarse correlated equilibrium (Moulin and Vial 1978), each player finds it optimal to commit ex ante to the future outcome from a probabilistic correlation device instead of playing any strategy of their own. In this paper, we consider a specific two-person game with unique pure Nash and correlated equilibrium and test the concept of coarse correlated equilibrium with a device which is an equally weighted lottery over three symmetric outcomes in the game including the Nash equilibrium, with higher expected payoff than the Nash payoff (as in Moulin and Vial 1978). We also test an individual choice between a lottery over the same payoffs with equal probabilities and the sure payoff as in the Nash equilibrium of the game. Subjects choose the individual lottery, however, they do not commit to the device in the game and instead coordinate to play the Nash equilibrium. We explain this behaviour as an equilibrium in the game

Correlated Equilibrium Under Costly Disobedience

In this paper, we extend Aumann's (1974) well-known solution of correlated equilibrium to allow for a cost of disobedience for each player. Calling the new solution costly correlated equilibrium (CCE), we derive the necessary and sufficient conditions under which the set of CCE strictly expands when the players' cost of disobedience is increased by the mediator in any finite normal-form game. These conditions imply that for any game that has a Nash equilibrium (NE) that is unpure, the set of CCE strictly expands with the addition of even arbitrarily small cost of disobedience, whereas for games that have a unique NE in pure strategies, the set of CCE stays the same unless the cost gets sufficiently high. We also study the welfare implications and changes in the value of mediation with exogenous cost changes. We find that strictly better social outcomes can be attained and the value of mediation cannot decrease with an increase in the cost level. We also illustrate how our model can be integrated with a cost-selection game where players non-cooperatively choose their costs of disobedience before mediation occurs. We show that there exist cost-selection games in which setting the cost of disobedience at zero is a strictly dominated strategy for each player as well as games this strategy becomes weakly dominant for everyone

On the enforcement value of soft correlated equilibrium for two-facility simple linear congestion games

Exact enforcement values (Ashlagi I, Monderer D and Tennenholz M (2008) Journal of Artificial Intelligence 33:575-613) of soft correlated equilibrium (Forgó F (2010) Mathematical Social Sciences 60:186-190) for non-decreasing and mixed two-facility simple linear congestion games (including n-person chicken and prisoners' dilemma games) are determined and found to be 1 and 2, respectively. For non-inreasing two-facility simple linear congestion games lower and upper bounds are given for the enforcement value. The upper bound 1,265625 is significantly better than the previously known 1,33