Trembling hand perfect equilibrium
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Trembling hand perfect equilibrium is a refinement of Nash Equilibrium due to Reinhard Selten. 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 we define a perturbed game. A perturbed game is a copy of a base game, with the restriction that only fully mixed strategies are allowed to be played. A fully mixed strategy is a mixed strategy where every pure strategy is played with non-zero probability. This is the "trembling hands" of the players; they sometimes play a different strategy than the one they intended to play. Then we 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.
Example
The game represented in the following normal form matrix has two Nash equilibria, namely
Left Right Top 1, 1 2, 0 Bottom 0, 2 2, 2 Assume player 1 is playing a mixed strategy [(1-\epsilon, \epsilon)]. Player 2's expected payoff from playing L is:
- [1(1-\epsilon) + 2\epsilon = 1+\epsilon]
- [0(1-\epsilon) + 2\epsilon = 2\epsilon]
However, similar analysis fails for the strategy profile
Assume player 1 is playing a mixed strategy [(\epsilon, 1-\epsilon)]. Player 2's expected payoff from playing L is:
- [1\epsilon + 2(1-\epsilon) = 2-\epsilon]
- [0(\epsilon) + 2(1-\epsilon) = 2-2\epsilon]
Trembling hand perfect equilibria of two-player games
For two-player games, the set of trembling hand perfect equilibria coincides with the set of admissible equilibria, i.e., equilibria consisting of two undominated strategies. In the example above, we see that the imperfect equilibrium
< B,R> is not admissible, as L (weakly) dominates R for Player 2.Trembling hand perfection of equilibria of 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 prefect 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.
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.
References
Reinhard Selten. "A reexamination of the perfectness concept for equilibrium points in extensive games". International Journal of Game Theory 4:25--55, 1975.
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