Question

Argue that a strategy in a pure strategy Nash equilibrium must be (normal form) rationalizable.

Answer 1

In a pure strategy, players adopt a strategy that provides the best payoffs. In other words, a pure stretegy is the one that provides maximum profit or the best outcome to players. Therefore, it is regarded as the best strategy for every player of the game.

A pure Nash equilibrium is a specification of a strategy for each player such that no player would benefit by changing his strategy, provided the other players don't change their stratigies.

If anything is rationalizable, it means that it can be explained with logical reasons. This means that if a strategy leads to pure strategy Nash Equilibrium than the strategy must be explained with the logical reasins that why did the player adopted that strategy.

In a Pure Strategy Nash Equilibrium every player adopts that strategy which gives his best payoff or which gives him the most profit. Since everyone in this world thinks of his own profit (which is reasonable because if everyone maximizes his profit the overall profit of the society will be maximized) and the player in the game is also thinking of his profit and than adopting the strategy according to it. So the strategy is surely rationalizable.

All the strategies adopted by all the players in a game of Pure strategy Nash Equilibrium must be rationalizable because if any one strategy is not rationalizable than there is no Nash Equilibrium as the movement of that player from one strategy to another will benefit him, this is the violation of the condition of Nash Equilibrium and hence there is no Nash Equilibrium without rationalizable strategy.