1Decision rules in Game Theory

The concept of decision rule was used by J. P. Aubin in [1], [2], then developed by D. Carfì in [3] and [4] and other papers and monographs.

1.1Strategic base of a two player game

Definition (strategy base and bistrategy space). Let $(E,F)$ be a pair of non-empty sets, we call it strategy base of a two-player game. The first set $E$ is said the first player's strategy set; the second set $F$ is said the second player's strategy set. Any element $x$ of $E$ is said a first player's strategy and any element $y$ in $F$ is said a second player's strategy. Every pair of strategies $(x,y)\in E\times F$ is said a bistrategy of the strategy base $(E,F)$ and the cartesian product $E\times F$ is said the bistrategy space of the base $(E,F)$.


Interpretation and terminology. We call the two players of a game Emil and Frances: Emil, simply, stands for "first player"; Frances stands for "second player". Emil's aim is to choose a strategy $x$ in the set $E$ , Frances' aim is to choose a strategy $y$ in $F$, in order to form a bistrategy $(x,y)$ such that the strategy $x$ is an Emil's good response to the Frances' strategy $y$ and vice versa.

1.2Definition of decision rule

Definition (decision rule). Let $(E,F)$ be a strategy base of a two-player game. An Emil's decision rule on the base $(E,F)$ is a correspondence from $F$ to $E$, say $e:F\rightarrow E.$ Symmetrically, a Frances' decision rule on the base $(E,F)$ is a correspondence from $E$ to $F$, say $f:E\rightarrow F$.


The best reply reaction behavior in normal-form games, which appears in the definition of Nash equilibrium, can be viewed as a particular kind of decision rule.