On the Cost of Game Playing: How to Control the Expenses in Mixed Strategies

Abstract

Game theory typically assumes rational behavior of the players when looking for optimal solutions. Still in case of a mixed equilibrium, it allows players to choose any strategy from the mix in each repetition of the game as long as the optimal frequencies are met in the long run. Which strategy is chosen in a specific round may not be purely random but also depend on what strategy has just been played.

In many cases, playing a particular strategy is tied to cost or efforts. For instance, adding a new defensive strategy (e.g., applying a new virus scanner) requires some investment (implementation cost), but playing the strategy may incur some efforts as well (playing cost: a virus scan takes time and consumes resources, so too frequent scanning appears undesirable). If a security system successfully repels an attack, the attacker is most likely coming back using a different attack vector. Thus we here study repeated games in order to respond to changing attacks.

The effort to play a strategy may be quite dependent on what has been played before, and the switch from the last strategy to the new one, in the next instance (repetition) of the game, may come at what we call a switching cost. These can create an incentive to not play a certain mixed strategy. In cases when there equilibrium is unique, a player may have an incentive to nonetheless deviate from it to save costs, and thus gain more (only in a different way). So, the strategy plan should depend on the equilibrium and the (switching) cost for playing it.

The matter is essentially more complex than only asking for how to play a mixed strategy most efficiently; instead, we need to incorporate the switching costs into the game as an additional goal to be optimized. Those costs are indeed dependent on the equilibrium of the game itself. Thus, the usual dependency of the equilibrium on the payoffs is herein augmented by the converse dependency of the payoffs on the equilibrium. To handle this circular dependency, we will apply a generalized game-theoretic model that allows payoffs to be random variables (rather than real numbers). We show how to solve this new form of game and illustrate the method with an example.