Best answer

In game theory which is best answer (English best response ) of a player on the strategies of the other players that strategy that gives him the highest payout. The set of the best answers plays an important role in determining Nash equilibria .

Mathematical definition

In the following, denote the set of strategies of the player and be an element of this set, ie a strategy of the player . Furthermore, denote a combination of the strategies of players and the payoff function of the player . Be a normal form game. The set of player 1's best answers to player 2's strategy is defined as: ${\displaystyle S_{i}}$ ${\displaystyle i}$ ${\displaystyle s_{i}}$ ${\displaystyle i}$ ${\displaystyle (s_{1},\dotsc ,s_{n})}$ ${\displaystyle n}$ ${\displaystyle u_{i}(s_{1},\dotsc ,s_{n})}$ ${\displaystyle i}$ ${\displaystyle G=\{S_{1},S_{2};u_{1},u_{2}\}}$ ${\displaystyle BR_{1}(t)}$ ${\displaystyle t\in S_{2}}$

${\displaystyle BR_{1}(t)=\{s\in S_{1}\mid u_{1}(s,t)\geq u_{1}(s',t)\;\forall \;s'\in S_{1}\}}$

The same applies to player 2

${\displaystyle BR_{2}(s)=\{t\in S_{2}\mid u_{2}(s,t)\geq u_{2}(s,t')\;\ \forall \;\ t'\in S_{2}\}}$

Connection with the Nash equilibrium

The pair is a Nash equilibrium when both strategies are best answers to each other. So if: ${\displaystyle (s,t)}$

${\displaystyle s\in BR_{1}(t)}$ and ${\displaystyle t\in BR_{2}(s)}$

Best-answer correspondence

Matching pennies

Matching Pennies : This diagram shows the reaction correspondences and graphs . The reaction correspondences are functions if . The only Nash equilibrium in mixed strategies can be found at the intersection B of the two best-answer correspondences (the lines are shown with dashed lines so as not to be reminiscent of a swastika ). ${\displaystyle BR_{1}(q_{2}):=\arg \max\{\operatorname {E} [U(q_{1},q_{2})]\mid q_{2}\in Q_{1}\}}$ ${\displaystyle BR_{2}(q_{1}):=\arg \max\{\operatorname {E} [U(q_{1},q_{2})]\mid q_{2}\in Q_{2}\}}$ ${\displaystyle q_{i}\neq {\tfrac {1}{2}}}$

A famous decision problem in game theory is the game of matching pennies : two players put a coin on the table at the same time. If both coins are heads (K) or both coins are tails (Z), the two coins belong to player 1; if the two coins show different sides, then the two coins belong to player 2. Since the winner wins the loser's coin, it is a zero-sum game. The following representation results as a bimatrix :

Payout matrix for player 1 and player 2

	head	number
head	1 , −1	−1 , 1
number	−1 , 1	1 , −1

The following matrix is ​​obtained in the simplified representation:

${\displaystyle {\begin{array}{c|rr}&K&Z\\\hline K&1&-1\\Z&-1&1\end{array}}\quad {\text{also die Matrix}}\quad A=\left({\begin{array}{rr}1&-1\\-1&1\end{array}}\right)}$

Individual evidence

1. ↑ Wolfgang Leiniger: Introduction to game theory . P. 21.
2. ^ Jürgen Eichberger: Fundamentals of microeconomics . 2004, p. 420.