trembling hand perfect Nash equilibrium

Definition

Given a perturbation vector $\varepsilon$ , denote the maximum perturbation in ε-perturbed game $\Gamma(\varepsilon)$ as $M(\varepsilon) := \max_{i \in N, s_i \in S_i} \varepsilon_i(s_i)$ , and minimum perturbation as $m(\varepsilon) := \min_{i \in N, s_i \in S_i} \varepsilon_i(s_i)$ where $m(\varepsilon) > 0$ .

A mixed strategy vector $\sigma$ in a strategy-form game is a perfect equilibrium if there exists a sequence of perturbation vectors $(\varepsilon^k)_{k \in \mathbb{N}}$ satisfying $\lim_{k \to \infty} M(\varepsilon^k) = 0$ , and for each $k \in \mathbb{N}$ there exists an equilibrium $\sigma^k$ of $\Gamma(\varepsilon^k)$ such that

\lim_{k \to \infty} \sigma^k = \sigma

(A mixed strategy vector that is the limit of equilibria in perturbed games, where pertubation vectors are all positive, and converge to zero, is called a perfect equilibria.)

Definition

A Nash equilibrium $\hat \gamma$ of a game $\Gamma_N = \{\mathcal{N},\{\Delta(A_i)\},\{u_i\}\}$ is (normal form) trembling hand perfect if there is a sequence of perturbed games $\{\Gamma_\varepsilon\}_{k=1}^\infty$ ,

\begin{aligned} \Gamma_\varepsilon & = \{\mathcal{N},\{\Delta_\varepsilon(A_i)\},\{u_i(\cdot)\}\} \\ \Delta_\varepsilon(A_i) & = \{p_i : p_i(a_i) \geq \varepsilon_i(a_i) \text{ for all } a_i \in A_i \text{ and } \sum_{a_i \in A_i} p_i(a_i) = 1\} \\ \text{with } & \sum_{a_i \in A_i} \varepsilon_i(a_i) < 1 \end{aligned}

converges to $\Gamma_N$ , for which there is an associated sequence of Nash equilibria $\{p^k\}_{k=1}^\infty$ ( $\{\hat{\gamma}^k\}_{k=1}^\infty$ ) that converges to $p$ (or $\hat{\gamma}$ ).

Remarks

Myerson's definition - proper, robustness?
Selten
"mild" way
"stronger" version?
"robust" to payoffs, or pertubations
other concepts of robustness

Example

	L	R
U	1,1	2,0
D	0,2	2,2
for this [[normal-form game	normal form]] matrix game, only UL is trembling hand perfect (despite RD also being a NE)
(player 1 as row player, player 2 as column player)

Theorem (weakly dominated strategies)

In every perfect equilibrium, every (weakly) dominated strategy is chosen with probability zero.

(recall that weakly dominated strategy refers to strategy that is never better and sometimes worse)

Theorem (completely mixed strategies)

Every equilibrium in completely mixed strategies in a strategic-form game is a perfect equilibrium.

Notes

extensive-form trembling hand perfect equilibrium is a subset of subgame perfect equilibrium

References

T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999, p. 129.
M. Maschler, E. Solan, and S. Zamir, Game Theory, 1st ed. Cambridge University Press, 2013, pp. 266-267. doi: 10.1017/CBO9780511794216.
R. Selten, “Reexamination of the perfectness concept for equilibrium points in extensive games,” Int J Game Theory, vol. 4, no. 1, pp. 25–55, Mar. 1975, doi: 10.1007/BF01766400.
R. B. Myerson, “Refinements of the Nash equilibrium concept,” Int J Game Theory, vol. 7, no. 2, pp. 73–80, Jun. 1978, doi: 10.1007/BF01753236.
https://en.wikipedia.org/wiki/Trembling_hand_perfect_equilibrium