Events at CERGE-EI

Prof. Venkataraman Bhaskar

University of Texas at Austin, United States

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Authors: V Bhaskar, Maxwell B. Stinchcombe

Abstract: Refinements of Nash equilibrium hinge on the question: what inferences does a player draw about her opponent’s future behavior or his type at an information set that has zero prior probability under the equilibrium. We address this question by adding shocks to the payoffs of each player i at every terminal node w that are independent across players and across nodes, that have a sufficiently large support, and have a continuous distribution F^iw. A strategy profile s of the unperturbed game is:

• Purifiable if there exists some sequence of distributions F_n := (F_n^iw)i∈I, w∈W, converging weakly to the Dirac measures on 0, with a sequence of equilibria σ_n whose aggregates s_n converge to s.
• Strongly purifiable if for every sequence of distributions F_n , that converge weakly to the Dirac measures on 0, there exists a sequence of equilibria σn whose aggregates s_n converge to s.

First we consider finite games of perfect information with generic payoffs, with a unique backwards induction (BI) strategy profile. If each player moves at most once along any path, then the BI strategy profile is the unique purifiable profile and is strongly purifiable. However, if a player a player moves more than once along a path, as in the centipede game, then there exist purifiable Nash equilibria that are not subgame perfect. Moreover, the BI strategy profile is not purifiable for some sequences of payoff shocks. However, every purifiable profile induces the backwards induction outcome.

Next, we consider signaling games. In the beer-quiche game, both pooling equilibria (pooling on beer and pooling on quiche) are purifiable but not strongly purifiable. This leads us to investigate the implications of symmetric purifiability, where the payoff shocks of a player have the same distribution at every terminal node. We show that this selects the intuitive equilibrium (pooling on beer, which is symmetrically purifiable), since pooling on quiche is not symmetrically purifiable. We also show that purification cannot justify forward induction arguments, since equilibria that fail forward induction are strongly purifiable.