Best response

In game theory, the best response is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players' strategies as given. The concept of a best response is central to John Nash's most well-known theory, the Nash equilibrium which is dependent on each player selecting the best response during each period.

Best response correspondence

Fig. 1 - Reaction correspondence for player Y in the Stag Hunt game.

Best response correspondences (often represented by $b(\cdot )$) are used in the proof of the existence of mixed strategy Nash equilibria. For each player, one constructs a correspondence from the set of opponent strategy profiles into the set of the player's strategies. So, for any given set of opponent's strategies $\sigma _{-i}$, $b_{i}(\sigma _{-i})$ represents player i's best responses to $\sigma _{-i}$.

Fig. 2 - Reaction correspondence for player XY in the Stag Hunt game.

Response correspondences for all 2x2 normal form games can be drawn with a line for each player in a unit square. For example, in Fig.1 the dotted line shows the optimal probability that player Y plays 'Stag' (in the y-axis), as a function of the probability that player X plays Stag (shown in the x-axis). In Fig.2 the dotted line shows the optimal probability that player X plays 'Stag' (shown in the x-axis), as a function of the probability that player Y plays Stag (shown in the y-axis). Note that Fig.2 plots the independent and response variables in the opposite axes to those normally used, so that it may be superimposed onto the previous graph, to show the Nash equilibria at the points where the two player's best responses agree in Fig.3.

Fig.3 - Reaction correspondence for both players in the Stag Hunt game. Nash equilibria shown with points, where the two player's correspondences agree, ie. cross

Coordination games

Games in which players score highest when both players choose the same strategy, such as the Stag hunt and Battle of the sexes games will have reaction correspondences which have the same shape as in Fig.3, where there is one Nash equilibrium in the bottom left corner, another in the top right, and a mixing Nash somewhere along the diagonal between the other two.

Discoordination games

Games such as the Game of chicken and Hawk-dove game in which players score highest when they choose opposite strategies, ie discoordinate, have reaction correspondences that cross in the opposite direction. Fig.4 shows the reaction correspondences for players in discoordination games. In the case of discoordination games there are three Nash equilibria, one in each of the top left and bottom right corners, where one player chooses one strategy, the other player chooses the opposite strategy. The third Nash equilibrium is a mixing strategy which lies somewhere along the diagonal from the bottom left to top right corners. If the players do not know which one of them is which, then the mixing Nash is an ESS, as play is confined to the bottom left to top right diagonal line. Otherwise an uncorrelated asymmetry is said to exist, and the corner Nash equilibria are ESSes.

Fig.4 - Reaction correspondence for both players in the Hawk-Dove game. Nash equilibria shown with points, where the two player's correspondences agree, ie. cross

Games with dominated strategies

Fig. 5 - Reaction correspondence for a game with a dominated strategy.

Games with dominated strategies have reaction correspondences which only cross at one point, which will be in either the bottom left, or top right corner in payoff symmetric 2x2 games. For instance, in the single-play Prisoner's dilemma, the Cooperate move is not optimal for any probability of opponent Cooperation. Fig.5 shows the reaction correspondence for such a game, where the dimensions are "Probability play Cooperate", the Nash equilibrium is in the lower left corner where neither player plays Cooperate. If the dimensions were defined as "Probability play Defect", then both players best response curves would be 1 for all opponent strategy probabilities and the reaction correspondences would cross (and form a Nash equilibrium) at the top right corner.

Other games

The three best response mappings above: coordination, discoordination and dominated strategy, are the only three possible mapping shapes for symmetrical 2x2 games (with the exception of the trivial fourth case in which payoffs are always equal for both moves). Any payoff symmetric 2x2 game will take one of these forms.

A wider range of reaction correspondences shapes is possible in 2x2 games with payoff asymmetries. For each player there are five possible best response shapes, shown in Fig.6. From left to right these are: dominated strategy (always play 2), dominated strategy (always play 1), rising (play strategy 2 if probability that the other player plays 2 is above threshold), falling (play strategy 1 if probability that the other player plays 2 is above threshold), and indifferent (both strategies play equally well under all conditions).

Fig. 6 - The five possible reaction correspondences for a player in a 2x2 game., The axes are assumed to show the probability that the player plays their strategy 1. From left to right: A) Always play 2, strategy 1 is dominated, B) Always play 1, strategy 2 is dominated, C) Strategy 1 best when opponent plays his strategy 1 and 2 best when opponent plays his 2, D) Strategy 1 best when opponent plays his strategy 2 and 2 best when opponent plays his 1, E) Both strategies play equally well no matter what the opponent plays.

While there are only four possible types of payoff symmetric 2x2 games (of which one is trivial), the five different best response curves per player allow for a larger number of payoff asymmetric game types. Many of these are not truly different from each other. The dimensions may be redefined (exchange names of strategies 1 and 2) to produce symmetrical games which are logically identical.

Matching Pennies

One well-known game with payoff asymmetries is the Matching pennies game. In this game one player, the row player - graphed on the y dimension, wins if the players coordinate (both choose heads or both choose tails) while the other player, the column player - shown in the x-axis, wins if the players discoorinate. Player Y's reaction correspondence is that of a coordination game, while that of player X is a discoordination game. The mixing Nash equilibrium is an evolutionarily stable strategy.

Fig. 7 - Reaction correspondences for players in the matching pennies game. The leftmost mapping is for the coordinating player, the middle shows the mapping for the discoordinating player. The sole Nash equilibrium, an evolutionarily stable strategy is shown in the right hand graph.

Best response dynamics

In evolutionary game theory, best response dynamics represents a class of strategy updating rules, where players strategies in the next round are determined by their best responses to some subset of the population. Some examples include:

• In a large population model, players choose their next action probabilistically based on which strategies are best responses to the population as a whole.
• In a spatial model, players choose (in the next round) the action that is the best response to all of their neighbors (see Elison 1993).

Importantly, in these models players only choose the best response on the next round that would give them the highest payoff on the next round. Players do not consider the effect that choosing a strategy on the next round would have on future play in the game. This constraint results in the dynamical rule often being called myopic best response.

References

• Ellison, G. (1993) "Learning, Local Interaction, and Coordination" Econometrica 61: 1047-1071
• Gibbons, R. (1992) "A primer in game theory" (pp. 33-49) Harvester-Wheatsheaf.