Definition (perturbation vector)

Let Γ=(N,(Si)i∈N,(ui)i∈N)\Gamma = (N, (S_i)_{i \in N}, (u_i)_{i \in N}) be a game in strategic-form in which the set of pure strategies of each player is finite. A perturbation vector of player ii is a vector εi=(εi(si))si∈Si\varepsilon_i = (\varepsilon_i(s_i))_{s_i \in S_i} satisfying εi(si)>0\varepsilon_i(s_i) > 0 for each si∈Sis_i \in S_i, and

∑si∈Siεi(si)≤1,∀i∈N\sum_{s_i \in S_i} \varepsilon_i (s_i) \leq 1, \quad \forall i \in N

A perturbation vector is a vector ε=(εi)i∈N\varepsilon = (\varepsilon_i)_{i \in N}.

maximum and minimum perturbation

Denote the maximum perturbation in ε\varepsilon-perturbed game Γ(ε)\Gamma(\varepsilon) as M(ε):=maxi∈N,si∈Si⁡εi(si)M(\varepsilon) := \max_{i \in N, s_i \in S_i} \varepsilon_i(s_i), and minimum perturbation as m(ε):=mini∈N,si∈Si⁡εi(si)m(\varepsilon) := \min_{i \in N, s_i \in S_i} \varepsilon_i(s_i) where m(ε)>0m(\varepsilon) > 0.

Definition (ε\varepsilon-perturbed game)

For each perturbation vector ε\varepsilon, the ε\varepsilon-perturbed game is the game Γ(ε)=(N,(Σi(εi))i∈N,(ui)i∈N)\Gamma(\varepsilon) = (N, (\Sigma_i(\varepsilon_i))_{i \in N}, (u_i)_{i \in N}) where player ii's strategy set is

Σi(εi):={σi∈Σi:σi(si)≥εi(si),∀si∈Si}\Sigma_i(\varepsilon_i) := \{ \sigma_i \in \Sigma_i : \sigma_i(s_i) \geq \varepsilon_i(s_i), \quad \forall s_i \in S_i \}

Notes

Intuition: assign a minimum for each pure strategy such that no pure strategy is explored with zero probability

See also

  • trembling hand

References

  1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 264.
  2. Jackson, Matthew O., Tomas Rodriguez-Barraquer, and Xu Tan. "Epsilon-equilibria of perturbed games." Games and Economic Behavior 75.1 (2012): 198-216. https://web.stanford.edu/~jacksonm/emailgame.pdf
  3. https://en.wikipedia.org/wiki/Trembling_hand_perfect_equilibrium