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 n {\displaystyle n}-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 σ = ( σ 1 , … , σ n ) {\displaystyle \sigma =(\sigma _{1},\ldots ,\sigma _{n})} as being trembling hand perfect if there is a sequence of perturbed games strategy profiles { σ k } k = 1 , 2 , … {\displaystyle \{\sigma ^{k}\}_{k=1,2,\ldots }} that converges to σ {\displaystyle \sigma } such that for every k {\displaystyle k} and every player 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} the strategy σ i {\displaystyle \sigma _{i}} is the best reply to σ − i k {\displaystyle \sigma _{-i}^{k}}.

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 ⟨ Up , Left ⟩ {\displaystyle \langle {\text{Up}},{\text{Left}}\rangle } and ⟨ Down , Right ⟩ {\displaystyle \langle {\text{Down}},{\text{Right}}\rangle }. However, only ⟨ U , L ⟩ {\displaystyle \langle {\text{U}},{\text{L}}\rangle } 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 ( 1 − ε , ε ) {\displaystyle (1-\varepsilon ,\varepsilon )}, for 0 < ε < 1 {\displaystyle 0<\varepsilon <1}.

Player 2's expected payoff from playing L is:

1 ( 1 − ε ) + 2 ε = 1 + ε {\displaystyle 1(1-\varepsilon )+2\varepsilon =1+\varepsilon }

Player 2's expected payoff from playing the strategy R is:

0 ( 1 − ε ) + 2 ε = 2 ε {\displaystyle 0(1-\varepsilon )+2\varepsilon =2\varepsilon }

For small values of ε {\displaystyle \varepsilon }, 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 ( 1 − ε , ε ) {\displaystyle (1-\varepsilon ,\varepsilon )}. Hence ⟨ U , L ⟩ {\displaystyle \langle {\text{U}},{\text{L}}\rangle } is trembling-hand perfect.

However, a similar analysis fails for the strategy profile ⟨ D , R ⟩ {\displaystyle \langle {\text{D}},{\text{R}}\rangle }.

Assume player 2 is playing a mixed strategy ( ε , 1 − ε ) {\displaystyle (\varepsilon ,1-\varepsilon )}. Player 1's expected payoff from playing U is:

1 ε + 2 ( 1 − ε ) = 2 − ε {\displaystyle 1\varepsilon +2(1-\varepsilon )=2-\varepsilon }

Player 1's expected payoff from playing D is:

0 ε + 2 ( 1 − ε ) = 2 − 2 ε {\displaystyle 0\varepsilon +2(1-\varepsilon )=2-2\varepsilon }

For all positive values of ε {\displaystyle \varepsilon }, player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U. Hence ⟨ D , R ⟩ {\displaystyle \langle {\text{D}},{\text{R}}\rangle } 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.[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) pointed out that perfection is sensitive to the addition of a strictly dominated strategy, and instead proposed another refinement, known as proper equilibrium.