Which Strategy to Choose?

Many of the world’s most outstanding theater plays derive their dramatic effect from the fact that the hero does not follow the path of action that corresponds to what the audience considers the dominant strategy. This holds for Brutus in Shakespeare’sShakespeareJulius CesarJulius Cesar as well as for Schiller’s Wallenstein, both characters who hesitate to grab power and thereby make use of the possibility to escape their fate. However, if players do not have dominant strategies and there are more than one Nash equilibriaNash equilibrium so that players control alternative equilibriumEquilibrium strategies, then the decision problem becomes even more challenging. Selten’sSelten, Reinhard trembling hand perfectness can be applied to select equilibrium strategies which are still adequate even when it is assumed that the other players deviate from their equilibrium strategies in the form of small ε trembles. Often this reduces the set of Nash equilibria and in some cases a singleton is left. In contrast to this operation, the concept of rationalizable strategiesRationalizable strategies leads to an expansion of the set of justifiable strategy choices. A strategy x is rationalizable if it is a best reply to strategy y and y is a best reply to a strategy z. In a Nash equilibriumNash equilibrium (x, y), x and y are mutually best replies. It is immediately understood that Nash equilibriumNash equilibrium strategies are rationalizable, but not all rationalizable strategiesRationalizable strategies are Nash equilibriumNash equilibrium strategies.