Question

what is the relationship between rationalizable strategies and those strategies that are never a best response?

Answer 1

Rationalizable strategies (of a player A, say) refer to the strategies which are best responses to strategies played by the other player (say, player B). So, with the assumption that the players are rational (in a game), they will play those strategies which are the best response strategies. This is because, these strategies give them a higher payoff compared to the other strategies.

The strategies which are not best responses to any other strategies played by the other player are called never best response strategies. A rational player will never play these strategies as they yield lesser payoffs. They would, thus, prefer to play from among the set of rationalizable strategies.

Let us try to get the idea clear with an example.

Suppose there are 2 players---player 1 and player 2.

The strategy set of player 1 is given as {a₁ , a₂ , a₃ , a₄} and the strategy set of player 2 is given as {b₁ , b₂ , b₃ , b₄}.

The payoffs of the game are given as follows in the normal form:     

Let us find the set of best responses for player 2 first. For finding this, we assume that player 1 plays a particular strategy.

If player 1 plays a₁ , b₁ is player 2's best response.

If player 1 plays a₂ , b₂ is player 2's best response.

If player 1 plays a₃ , b₃ is player 2's best response.

Now, note that player 1 will never play a strategy like a₄ , as the payoffs from this strategy are no greater than what he would have gotten if he played a₁ , a₂ and a₃ .  

Hence a₄ is a never best response strategy of player 1 . His set of best response strategies (or rationalizable strategies) are : {a₁ , a₂ and a₃ }

Now, put your attention to player 2. How is he reacting? He plays b₁ , b₂ and b_3. So these are his best response strategies or rationalizable strategies. He never plays a strategy like b_4. So it becomes a never best response strategy for player 2.

While finding a Nash Equilibrium for a game, we can eliminate these never best response strategies to come to a truncated version of the original game. From there, we can argue which one can be the possible Nash Equilibrium of the game in question.