Chess-Like Games May Have No Uniform Nash Equilibria Even in Mixed Strategies

Figure 1

Two Chess-like game structures 𝒢₁ and 𝒢₂. In 𝒢₁, there are 3 players controlling one position each (i.e., 𝒢₁ is play-once), while in 𝒢₂ there are only two players who alternate turns; hence, each of them controls 3 positions: 𝑣₁,𝑣₃,𝑣₅ are controlled by player 1 and 𝑣₂,𝑣₄,𝑣₆ by player 2. In each position 𝑣_ℓ, the corresponding player has only two options: (𝑝) to proceed to 𝑣_ℓ+1 and (𝑡) to terminate at 𝑎_ℓ, where ℓ∈{1,2,3} (𝑣₄=𝑣₁) in 𝒢₁ and ℓ∈{1,…,6} (𝑣₇=𝑣₁) in 𝒢₂. To save space, we show only symbols 𝑎_ℓ, while the corresponding vertex names are omitted.