(Normal form) trembling hand perfect equilibrium | |
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A solution concept in game theory | |
Relationship | |
Subset of | Nash Equilibrium |
Superset of | Proper equilibrium |
Significance | |
Proposed by | Reinhard Selten |
In game theory, trembling hand perfect equilibrium is a refinement of Nash equilibrium due to Reinhard Selten.^{[1]} A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability.
Definition
First define a perturbed game. A perturbed game is a copy of a base game, with the restriction that only totally mixed strategies are allowed to be played. A totally mixed strategy is a mixed strategy where every strategy (both pure and mixed) is played with non-zero probability. This is the "trembling hands" of the players; they sometimes play a different strategy, other than the one they intended to play. Then define a strategy set S (in a base game) as being trembling hand perfect if there is a sequence of perturbed games that converge to the base game in which there is a series of Nash equilibria that converge to S.
Note: All completely mixed Nash equilibria are perfect.
Note 2: The mixed strategy extension of any finite normal-form game has at least one perfect equilibrium.^{[2]}
Example
The game represented in the following normal form matrix has two pure strategy Nash equilibria, namely and . However, only is trembling-hand perfect.
Left | Right | |
Up | 1, 1 | 2, 0 |
Down | 0, 2 | 2, 2 |
Trembling hand perfect equilibrium |
Assume player 1 (the row player) is playing a mixed strategy , for .
Player 2's expected payoff from playing L is:
Player 2's expected payoff from playing the strategy R is:
For small values of , player 2 maximizes his expected payoff by placing a minimal weight on R and maximal weight on L. By symmetry, player 1 should place a minimal weight on D and maximal weight on U if player 2 is playing the mixed strategy . Hence is trembling-hand perfect.
However, similar analysis fails for the strategy profile .
Assume player 2 is playing a mixed strategy . Player 1's expected payoff from playing U is:
Player 1's expected payoff from playing D is:
For all positive values of , player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U. Hence is not trembling-hand perfect because player 2 (and, by symmetry, player 1) maximizes his expected payoff by deviating most often to L if there is a small chance of error in the behavior of player 1.
Equilibria of two-player games
For 2x2 games, the set of trembling-hand perfect equilibria coincides with the set of equilibria consisting of two undominated strategies. In the example above, we see that the equilibrium <Down,Right> is imperfect, as Left (weakly) dominates Right for Player 2 and Up (weakly) dominates Down for Player 1.^{[3]}
Equilibria of extensive form games
Extensive-form trembling hand perfect equilibrium | |
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A solution concept in game theory | |
Relationship | |
Subset of | Subgame perfect equilibrium, Perfect Bayesian equilibrium, Sequential equilibrium |
Significance | |
Proposed by | Reinhard Selten |
Used for | Extensive form games |
There are two possible ways of extending the definition of trembling hand perfection to extensive form games.
- One may interpret the extensive form as being merely a concise description of a normal form game and apply the concepts described above to this normal form game. In the resulting perturbed games, every strategy of the extensive-form game must be played with non-zero probability. This leads to the notion of a normal-form trembling hand perfect equilibrium.
- Alternatively, one may recall that trembles are to be interpreted as modelling mistakes made by the players with some negligible probability when the game is played. Such a mistake would most likely consist of a player making another move than the one intended at some point during play. It would hardly consist of the player choosing another strategy than intended, i.e. a wrong plan for playing the entire game. To capture this, one may define the perturbed game by requiring that every move at every information set is taken with non-zero probability. Limits of equilibria of such perturbed games as the tremble probabilities goes to zero are called extensive-form trembling hand perfect equilibria.
The notions of normal-form and extensive-form trembling hand perfect equilibria are incomparable, i.e., an equilibrium of an extensive-form game may be normal-form trembling hand perfect but not extensive-form trembling hand perfect and vice versa. As an extreme example of this, Jean-François Mertens has given an example of a two-player extensive form game where no extensive-form trembling hand perfect equilibrium is admissible, i.e., the sets of extensive-form and normal-form trembling hand perfect equilibria for this game are disjoint.^{[citation needed]}
An extensive-form trembling hand perfect equilibrium is also a sequential equilibrium. A normal-form trembling hand perfect equilibrium of an extensive form game may be sequential but is not necessarily so. In fact, a normal-form trembling hand perfect equilibrium does not even have to be subgame perfect.
Problems with perfection
Myerson (1978)^{[4]} pointed out that perfection is sensitive to the addition of a strictly dominated strategy, and instead proposed another refinement, known as proper equilibrium.
References
- ^ Selten, R. (1975). "A Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games". International Journal of Game Theory. 4 (1): 25–55. doi:10.1007/BF01766400.
- ^ Selten, R.: Reexamination of the perfectness concept for equilibrium points in extensive games. Int. J. Game Theory4, 1975, 25–55.
- ^ Van Damme, Eric (1987). Stability and Perfection of Nash Equilibria. doi:10.1007/978-3-642-96978-2. ISBN 978-3-642-96980-5.
- ^ Myerson, Roger B. "Refinements of the Nash equilibrium concept." International journal of game theory 7.2 (1978): 73-80.
Further reading
- Osborne, Martin J.; Rubinstein, Ariel (1994). A Course in Game Theory. MIT Press. pp. 246–254. ISBN 9780262650403.