Keywords: Strategic form games
1. Preliminaries
A strategic form game is composed of the set of players $N$, a set of actions $A_i$, and a payoff function $u_i: A \xrightarrow{} \mathbb{R}$. Note that the payoff functions may differ among the players, but the domain of the function is the outcome space (the cartesian product of the actions set). The outcome of a game is the actions profile of each player.
1.1 The definition of the rational players
We just consider scenario where all the players are rational. The basic definition of rational is that the player has well-defined objectives over the set of the possible outcome, i.e. the payoff function should be a well-defined function whose domain is the outcome space.
Note that the assumption about rational players may not be a realistic one. An active study of lacking the rational players is called bounded rationality.
For a player in the strategic form game, if $u_i(a_1, \dots, a_n) > u_i(b_1, \dots, b_n)$, then we understand that player i like outcome $(a_1, \dots, a_n)$ strictly better than $(b_1, \dots, b_n)$
1.2 The formulation of the strategic form
We can formally define a game in strategic form as the tuple: $(N, \{A_i\}_{i \in N}, \{u_i\}_{i \in N})$.
More specifically, we sometimes consider the so-called symmetric game which follows two conditions:
a) $A_i = A_k$, $\forall i, k \in N$
b) $u_i(a_i, a_k, a_{-i, -k}) = u_k(a_k, a_i, a_{-i, -k})$.
Note that the second condition implies that the exchange of the actions will lead to the exchange of payoff. The second condition further indicate that the same actions of every two players will lead to the same payoff.
2. Solution concepts of strategic form game
Action $a_i \in A_i$ weakly dominate action $b_i \in A_i$ for player i if $u_i(a_i, a_{-i}) \ge u_i(b_i, a_{-i})$ for all $a_{-i} \in A_{-i}$, and strictly dominates action $b_i \in A$ for player i if $u_i(a_i, a_{-i}) > u_i(b_i, a_{-i})$. We say an action strictly dominate if for any other action $a$, $u_i(a_i, a_{-i}) > u_i(a, a_{-i})$. A rational player will not choose the dominated (被占优) action. We can further search for strictly dominated actions in this smaller game, that is, eliminate next the strictly dominated actions of another player requiring dominance only against actions not yet eliminated. The elimination process is called iterated elimination of strictly dominated actions (IESD).
Proposition: If both players have strictly dominant actions, then IESD actions leads to the unique dominant strategy equilibrium.
Definition: A strategic form game $G$ is dominance solvable if IESD actions leads to a unique outcome.
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