对称博弈

对于参与者${\displaystyle i}$ 而言，博弈的收益为${\displaystyle U_{i}\colon A_{1}\times A_{2}\times \cdots \times A_{n}\longrightarrow \mathbb {R} }$ ，其中${\displaystyle A_{i}}$ 为参与者${\displaystyle i}$ 的决策集合。如果存在${\displaystyle A_{1}=A_{2}=\ldots =A_{N}}$ ，那么对于任何排列${\displaystyle \pi }$ 而言该博弈为对称博弈。

${\displaystyle U_{\pi (i)}(a_{1},\ldots ,a_{i},\ldots ,a_{N})=U_{i}(a_{\pi (1)},\ldots ,a_{\pi (i)},\ldots ,a_{\pi (N)}).}$ ^[1]

帕萨·达斯古普塔和埃里克·马斯金给出了以下定义，此后，这一定义在经济学文献中反复出现

${\displaystyle U_{i}(a_{1},\ldots ,a_{i},\ldots ,a_{N})=U_{\pi (i)}(a_{\pi (1)},\ldots ,a_{\pi (i)},\ldots ,a_{\pi (N)}).}$