Correlated equilibria, good and bad : an experimental study

We report results from an experiment that explores the empirical validity of correlated equilibrium, an important generalization of the Nash equilibrium concept. Specifically, we seek to understand the conditions under which subjects playing the game of Chicken will condition their behavior on private, third–party recommendations drawn from known distributions. In a “good–recommendations” treatment, the distribution we use is a correlated equilibrium with payoffs better than any symmetric payoff in the convex hull of Nash equilibrium payoff vectors. In a “bad–recommendations” treatment, the distribution is a correlated equilibrium with payoffs worse than any Nash equilibrium payoff vector. In a “Nash–recommendations” treatment, the distribution is a convex combination of Nash equilibrium outcomes (which is also a correlated equilibrium), and in a fourth “very–good–recommendations” treatment, the distribution yields high payoffs, but is not a correlated equilibrium. We compare behavior in all of these treatments to the case where subjects do not receive recommendations. We find that when recommendations are not given to subjects, behavior is very close to mixed–strategy Nash equilibrium play. When recommendations are given, behavior does differ from mixed–strategy Nash equilibrium, with the nature of the differ- ences varying according to the treatment. Our main finding is that subjects will follow third–party recommendations only if those recommendations derive from a correlated equilibrium, and further,if that correlated equilibrium is payoff–enhancing relative to the available Nash equilibria

Efficiency in the Trust Game: an Experimental Study of Preplay Contracting

We use a human-subjects experiment to test the effects of a simple mechanism designed to increase cooperation and efficiency in the trust game. In the equilibrium of the standard trust game, the investor does not invest, foreseeing that the allocator would have kept all of the returns from investment. Our mechanism adds a preplay escrow stage, in which the allocator places an amount (possibly zero) into escrow, to be forfeited if he keeps the proceeds of investment for himself. In the experiment, we vary the amounts that can be put into escrow. Our baseline treatment has no escrow. In a second treatment, only low escrow choices are possible, so the equilibrium is unaffected. In our third treatment, there is an escrow amount high enough that, in equilibrium, investment and sharing of the proceeds will occur. Two additional treatments mirror our second and third, except that in these, the escrow amount is randomly chosen and imposed on the allocator. We find that the high escrow amount, when chosen, does lead to the predicted efficient result. Contrary to the equilibrium prediction, we also find substantial investment in both the baseline and “low-escrow” treatments, leading to markedly higher efficiency than predicted (albeit well below that when the high amount is chosen). Over time, however, cooperation and efficiency after low or zero escrow amounts decline. We find only weak evidence for “crowding-out”, which predicts that given a low or zero amount placed into escrow in non-baseline treatments, investment and efficiency would actually be lower than in the baseline. We also find that initially, investment is more likely after allocators place the maximum possible amount into escrow – as if this action by allocators is being (mistakenly) read by investors as a signal that allocators plan to share. All of these results are seen when escrow choices are imposed as well as when they are voluntary.experiment, trust game, incentives, signal, crowding out

Correlated Equilibria, Good and Bad: An Experimental Study

We report results from an experiment that explores the empirical validity of correlated equilibrium, an important generalization of the Nash equilibrium concept. Specifically, we seek to understand the conditions under which subjects will condition their behavior on private, third-party recommendations drawn from known distributions in playing the game of Chicken. In a `good-recommendations` treatment, the distribution is such that following recommendations comprises a correlated equilibrium with payoffs better than any symmetric payoff in the convex hull of Nash equilibrium payoff vectors. In a `bad-recommendations` treatment, the distribution is such that following recommendations comprises a correlated equilibrium with payoffs worse than any Nash equilibrium payoff vector. In a `Nash-recommendations` treatment, the distribution is a convex combination of Nash equilibrium outcomes (which is also a correlated equilibrium), and in a fourth `very-good-recommendations` treatment, the distribution yields high payoffs, but following recommendations does not comprise a correlated equilibrium. We compare behavior in all of these treatments to the case where subjects do not receive recommendations. We find that when recommendations are not given to subjects, behavior is very close to mixed-strategy Nash equilibrium play. When recommendations are given, behavior does differ from mixed-strategy Nash equilibrium, with the nature of the differences varying according to the treatment. Our main finding is that subjects will follow third-party recommendations only if those recommendations derive from a correlated equilibrium, and further, if that correlated equilibrium is payoff-enhancing relative to the available Nash equilibria.

Bargaining with random arbitration: an experimental study

We use a laboratory experiment to study bargaining in the presence of random arbitration. Two players make simultaneous demands; if compatible, each receives the amount demanded as in the standard Nash demand game. If bargainers’ demands are incompatible, then rather than bargainers receiving their disagreement payoffs with certainty, they receive them only with exogenous probability 1−q. With probability q, there is random arbitration instead, with one bargainer randomly selected to receive his/her demand and the other bargainer receiving the remainder. The bargaining set is asymmetric, with one bargainer favoured over the other. We set disagreement payoffs to zero, and vary q over several values ranging from zero to one. Our main experimental results support the directional predictions of standard game theory (though the success of its point predictions is mixed). In the spirit of typical results for conventional arbitration, we observe a strong chilling effect on bargaining for values of q near one, with extreme demands and low agreement rates in these treatments. For the most part, increases in q reinforce the built-in asymmetry of the game, further benefiting the favoured player at the expense of the unfavoured player. The effects we find are non-uniform in q: over some fairly large ranges, increases in q have minimal effect on bargaining outcomes, but for other values of q, a small additional increase in q leads to sharp changes in results.Nash demand game, random arbitration, chilling effect, equilibrium selection,experiment.

How sensitive are bargaining outcomes to changes in disagreement payoffs?

We use a human–subjects experiment to investigate how bargaining outcomes are affected by changes in bargainers’disagreement payoffs. Subjects bargain against changing opponents, with an asymmetric disagreement outcome that varies over plays of the game. Both bargaining parties are informed of both disagreement payoffs (and the cake size) prior to bargaining. We find that bargaining outcomes do vary with the disagreement outcome, but subjects severely under–react to changes in their own disagreement payoff and to changes in the opponent’s disagreement payoff, relative to the risk–neutral prediction. This effect is observed in a standard Nash demand game and a related unstructured bargaining game, and for two different cake sizes varying by a factor of four. We show theoretically that standard models of expected utility maximisation are unable to account for this under–responsiveness – even when risk aversion is introduced. We also show that other–regarding preferences can explain our main results.Nash demand game, unstructured bargaining, disagreement, experiment, risk aversion, social preference, other–regarding behaviour, bargaining power.

Signaling and Countersignaling: A Theory of Understatement

In signaling environments ranging from consumption to education, high quality senders often shun the standard signals that should separate them from lower quality senders. We find that allowing for additional, noisy information on sender quality permits equilibria where medium types signal to separate themselves from low types, but high types then choose to not signal or countersignal. High types not only save costs by relying on the additional information to stochastically separate them from low types, but countersignaling itself is a signal of confidence which separates high types from medium types. Experimental results confirm that subjects can learn to countersignal.signaling; countersignaling; understatement

Payoff levels, loss avoidance, and equilibrium selection in the Stag Hunt: an experimental study

Game theorists typically assume that changing a game’s payoff levels—by adding the same constant to, or subtracting it from, all payoffs—should not affect behavior. While this invariance is an implication of the theory when payoffs mirror expected utilities, it is an empirical question when the “payoffs” are actually money amounts. In particular, if individuals treat monetary gains and losses differently, then payoff–level changes may matter when they result in positive payoffs becoming negative, or vice versa. We report the results of a human–subjects experiment designed to test for two types of loss avoidance: certain–loss avoidance (avoiding a strategy leading to a sure loss, in favor of an alternative that might lead to a gain) and possible–loss avoidance (avoiding a strategy leading to a possible loss, in favor of an alternative that leads to a sure gain). Subjects in the experiment play three versions of Stag Hunt, which are identical up to the level of payoffs, under a variety of treatments. We find differences in behavior across the three versions of Stag Hunt; these differences are hard to detect in the first round of play, but grow over time. When significant, the differences we find are in the direction predicted by certain– and possible–loss avoidance. Our results carry implications for games with multiple equilibria, and for theories that attempt to select among equilibria in such games