Trembling hand perfect equilibrium
Trembling hand perfect equilibrium
In game-theory, trembling hand perfect equilibrium is a type of refinement of a nash-equilibrium that was first proposed by 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 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 in an <math>n</math>-player strategic game where every pure strategy is played with positive 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 mixed strategy profile <math>\sigma=(\sigma_1,\ldots,\sigma_n)</math> as being trembling hand perfect if there is a sequence of perturbed games strategy profiles <math>\{\sigma^k\}_{k=1,2,\ldots}</math> that converges to <math>\sigma</math> such that for every <math>k</math> and every player <math>1\leq i \leq n</math> the strategy <math>\sigma_i</math> is the best reply to <math>\sigma^k_{-i}</math>.
- 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.
Example The game represented in the following normal form matrix has two pure strategies Nash equilibria, namely <math>\langle \text{Up}, \text{Left}\rangle</math> and <math>\langle \text{Down}, \text{Right}\rangle</math>. However, only <math>\langle \text{U},\text{L}\rangle</math> is trembling-hand perfect.
Assume player 1 (the row player) is playing a mixed strategy <math>(1-\varepsilon, \varepsilon)</math>, for <math> 0<\varepsilon <1</math>.
Player 2's expected payoff from playing L is: :<math>1(1-\varepsilon) + 2\varepsilon = 1+\varepsilon</math>
Player 2's expected payoff from playing the strategy R is:
:<math>0(1-\varepsilon) + 2\varepsilon = 2\varepsilon</math>
For small values of <math>\varepsilon</math>, player 2 maximizes his expected payoff by placing a minimal weight on R and a maximal weight on L. By symmetry, player 1 should place a minimal weight on D and a maximal weight on U if player 2 is playing the mixed strategy <math>(1-\varepsilon, \varepsilon)</math>. Hence <math>\langle \text{U},\text{L}\rangle</math> is trembling-hand perfect.
However, a similar analysis fails for the strategy profile <math>\langle \text{D}, \text{R}\rangle</math>.
Assume player 2 is playing a mixed strategy <math>(\varepsilon, 1-\varepsilon)</math>. Player 1's expected payoff from playing U is:
:<math>1\varepsilon + 2(1-\varepsilon) = 2-\varepsilon</math>
Player 1's expected payoff from playing D is:
:<math>0\varepsilon + 2(1-\varepsilon) = 2-2\varepsilon</math>
For all positive values of <math>\varepsilon</math>, player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U. Hence <math>\langle \text{D}, \text{R}\rangle</math> 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 2×2 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.
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 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 go 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.
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.