Potential games

Abstract

These games are studied in D. Monderer and L.S. Shapley (1996). See also Voorneveld (1999). Let Γ = 〈X₁, X₂,…, X _p , K₁, K₂,…, K _p 〉 be a p-person game and let P: Π ^p_i=1 X _i → ℝ be a real-valued function on the Cartesian product of the strategy spaces of Γ. Then P is called a potential of Γ if for each i ∈ {1, 2,…, p}, each x⁻ⁱ ∈ Π _k≠i X _k and all x _i , x ^′ _i ∈ X _i we have

$${K_i}\left( {{x^{ - i}},{{x'}_i}} \right) - {K_i}\left( {{x^{ - i}},{x_i}} \right) = P\left( {{x^{ - i}},{{x'}_i}} \right) - P\left( {{x^{ - i}},{x_i}} \right)$$

.