11562

# Best Response in Static Two-Player Games

This Demonstration gives a visual interpretation of the best response in static two-player games using an example from a textbook on game theory [1, pp. 59]. The game is considered from the viewpoint of the first player, as it is easier to visualize strategies if an opponent is limited to just two strategies.

### DETAILS

The game structure panel gives a two-player game with payoff matrix. For simplicity, only the first player's utilities are shown, leaving the second player's parameters as placeholders; they do not play a decisive role in the analysis. The first player's strategies are the set , and the second player's strategies are .
The best response is determined by the beliefs of a player, which is a distribution of probabilities over the strategies of the other player [1, pp. 39].
If player 1 believes that player 2 will choose strategy with probability , then her best response will be determined by the red line. If , then the second player may play only pure strategies, or . The fact that gets any value within the interval reflects uncertainty of the first player about the behavior of the opponent.
In theory, player 1 should be guided only by personal beliefs in order to make a rational decision. Look closely at the best response function (thick red line). This is, in essence, the function that maps beliefs to strategies. Consider an example when lies between the values 0 and 1 with the initial setting . This means that if , then player 1 activates strategy ; if , then strategy obviously brings a larger payoff. Finally, at the vertex, the player may choose or . With the initial setting, there is no provision for the rational playing of strategy .
The best response is closely related to the concept of dominance. To consider an example, click the "initial settings" button. You can see that strategy is never a best response, which means it is strictly dominated by the combination of the other strategies, and . When you vary the sliders, the payoff matrix is changed for the first player, and so the game changes.
It is important to understand that a player forms a response taking into account her own beliefs concerning probabilities of the strategies of the opponent. Player payoffs determine the form of the best response. Concrete values of the utilities of the opponent, even if known to the player, may affect the beliefs but do not change the model.
Click the "BR" checkbox to see the best response of the first player in the game.
Reference
[1] J. Watson, Strategy: An Introduction to Game Theory, 2nd ed., New York: W. W. Norton, 2008.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.