Table 3 Required notations and basic notions of game theory in the present study
Notation | Description |
|---|---|
\(\lambda_{k}\) | The strategy of each player (supplier) k, where \(k = 1 2 \ldots K\) |
\(\lambda_{ - k} \equiv \left( {\lambda_{1} \ldots \lambda_{i - 1} \lambda_{i + 1} \ldots \lambda_{k} } \right)\) | A combined strategy of all players except the strategy of player k |
\(\lambda \equiv \left( {\lambda_{k} \lambda_{ - k} } \right)\) | The strategy profile of all players |
*\(\pi_{k} \left( s \right)\) | Player k’s payoff (profit in this study) associated with the strategy profile s |
Nash equilibrium | A strategy profile in which no player can benefit from deviating its current strategy unilaterally |
\(G\left( {x y} \right) \equiv cardinality of set:\left\{ {k \in \left\{ {1 \ldots K} \right\}{|}\pi_{k} \left( {y_{k} x_{ - k} } \right) \ge \pi_{k} \left( x \right) y_{k} \ne x_{k} } \right\}\) | The number of players who can benefit from playing \(y_{k}\) when everyone else plays \(x_{ - k}\). Note that y and x are two strategy profiles |
\(G\left( {y x} \right) \equiv cardinality offset:\left\{ {k \in \left\{ {1 \ldots K} \right\}{|}\pi_{k} \left( {x_{k} y_{ - k} } \right) \ge \pi_{k} \left( y \right) y_{k} \ne x_{k} } \right\}\) | The number of players who can benefit from playing \(x_{k}\) when everyone else plays \(y_{ - k}\). Note that y and x are two strategy profiles |