Abstract
In Aumann (Games Econ Behav 8(1):6–19, 1995, Games Econ Behav 23(1):97–105, 1998), time is assumed implicitly in the description of games of perfect information, and it is part of the epistemic distinction between ex-ante and ex-post knowledge. We show that ex-post knowledge in these papers can be expressed by ex-ante knowledge and therefore epistemically, time is irrelevant to the analysis. Furthermore, we show that material rationality by weak dominance and by expectation can be expressed in terms of the timeless strategic form of the game.
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Notes
Opinions are split on the importance of the extensive form for the analysis of a game. The strategic form and the extensive form of a game were considered by von Neumann and Morgenstern (1944, section 12.1.1) as “strictly equivalent”, and the choice of the form that should serve as the basis of the analysis of the game, a matter of convenience. The analysis of Kohlberg and Mertens (1986) also emphasized the dispensability of the extensive form of the game. Dalkey (1953), Thompson (1952) and Elmes and Reny (1994) studied transformations of the extensive form that preserve its strategic nature and result in the strategic form. Here we study this question in the epistemic setup.
Aumann (1998) showed that common knowledge of material rationality in the centipede game implies that the first player stops the game immediately, but did not characterize the profiles that are played under this assumption in general. This has been shown by Hillas and Samet (2014) to be the set of non-probabilistic correlated equilibria.
Weak dominance here is an epistemic notion. It refers to a strategy known to the player to be weakly inferior to another strategy. Such an inferior strategy is weakly dominated relative to the set of profiles of the player’s opponents that the player does not exclude.
The event \(\lnot X \cup Y\) corresponds to the the assertion that either X does not hold, or else Y holds. But it also correspond to the assertion that if X holds then Y holds. In logic, the ‘if...then...’ construction, in this sense, is called material implication.
We can read (5) alternatively as a conditional. If i does not exclude the possibility that v is reached (that is, if she does not know that v is not reached), then she does not know that if v is reached then \(t_i\) dominates \(\varvec{s}_i\) at v.
In the model of knowledge and belief that we use here, the measurability of \(\tau _i\) is tantamount to saying that each player knows her beliefs, and the condition \(\tau _i(\Pi _i(\omega ))=1\) is equivalent to saying that each player is certain of whatever she knows.
References
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Aumann RJ (1995) Backward induction and common knowledge of rationality. Games Econ Behav 8(1):6–19
Aumann RJ (1998) On the centipede game. Games Econ Behav 23(1):97–105
Dalkey N (1953) Eqnivalence of information patterns and essentially determinate garnes. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of garnes, vol 2. Princeton University Press, Princeton, pp 127–143
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Hillas J, Samet D (2014) Non-probabilistic correlated equilibrium as an expression of non-Bayesian rationality. http://www.tau.ac.il/~samet/papers/npce.pdf
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Thompson FB (1952) Equivalence of games in extensive form. RM 759, The Rand Corporation. Reprinted in Kuhn HW (ed) Classics in game theory, 1997. Princeton University Press, Princeton
von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton
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Financial support of Israel Science Foundation through Grant #1827/14 is acknowledged.
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Samet, D. On the dispensable role of time in games of perfect information. Int J Game Theory 45, 375–387 (2016). https://doi.org/10.1007/s00182-015-0510-x
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DOI: https://doi.org/10.1007/s00182-015-0510-x