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

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