I - Basic Results on Normal Form Games

Summary

Non-cooperative games (or strategic games) are mainly studied through two models: normal form or extensive form. The latter will be presented in Chapter II. The former describes the choice spaces of each player and the result of their common choices. This is evaluated in terms of the players' von Neumann–Morgenstern utilities (i.e., the utility of a random variable is its expected utility (von Neumann and Morgenstern, 1944, Chapter I, 3.5)), and hence the following definition:

A normal form game is defined by a set of I, strategy spaces Sⁱ, i ∈ I, and real payoff functionsFⁱ, i ∈ I, on S = Π_i Sⁱ.

It is finite (or a bi-matrix game if #I = 2) if I and all Sⁱ are finite.

Under suitable measurability conditions one defines the mixed extension of a game G = (I, (Sⁱ, Fⁱ)_i∈I) as the game Γ = (I, (Σⁱ, φⁱ)_i∈I), where Σⁱ is the set of probabilities on Sⁱ and φⁱ (σ) = ∫_SFⁱ(S) Π_i∈I σⁱ(dsⁱ). An element of Sⁱ will be called a pure strategy, while an element of Σⁱ will be called a mixed strategy (of player i). Unless explicitly specified (or self-evident), the following definitions are always used on the mixed extension of the game.

sⁱ is a dominant strategy of player i if Fⁱ(sⁱ, s⁻ⁱ) ≥ Fⁱ (tⁱ, s⁻ⁱ) for all tⁱ in Sⁱ, and s⁻ⁱ ∈ S⁻ⁱ = Π_h≠iS^h.

sⁱ is dominated (resp. strictly dominated) if there exists tⁱ with Fⁱ(tⁱ, s⁻ⁱ) > Fⁱ (sⁱ, s⁻ⁱ) for some s⁻ⁱ (resp. all s⁻ⁱ) and Fⁱ(tⁱ, ·) ≥ Fⁱ(sⁱ,·).

sⁱ is an (ε-)best reply to s⁻ⁱ if Fⁱ(sⁱ, s⁻ⁱ) ≥ Fⁱ(tⁱ, s⁻ⁱ) (−ε) for all tⁱ ∈ Sⁱ.

An (ε-)equilibrium is an I-tuple s such that for every i, sⁱ is an (ε-)best reply to s⁻ⁱ.