Bimatrix

Example of a bimatrix in the heads or tails game

In game theory , a bimatrix is the matrix representation of a two-person game in normal form . The name "Bimatrix" comes from the fact that games can be described in normal form by two matrices - a matrix that describes the payouts of player 1 and a matrix that describes the payouts of player 2. ${\ displaystyle A}$ ${\ displaystyle B}$

Player 1 is often referred to as the "line player" and player 2 is often referred to as the "column player".

General representation

{\ displaystyle y_ {1}}

{\ displaystyle y_ {2}}

{\ displaystyle x_ {1}}

${\ displaystyle A}$	${\ displaystyle a}$

${\ displaystyle C}$	${\ displaystyle b}$

{\ displaystyle x_ {2}}

${\ displaystyle B}$	${\ displaystyle c}$

${\ displaystyle D}$	${\ displaystyle d}$

For this two-person game, it is symmetrical if . The bimatrix can thus be represented as follows: ${\ displaystyle a = A, b = B, c = C, d = D}$

{\ displaystyle y_ {1}}

{\ displaystyle y_ {2}}

{\ displaystyle x_ {1}}

${\ displaystyle A}$	${\ displaystyle A}$

${\ displaystyle C}$	${\ displaystyle B}$

{\ displaystyle x_ {2}}

${\ displaystyle B}$	${\ displaystyle C}$

${\ displaystyle D}$	${\ displaystyle D}$

If this game is true, then it is a prisoner's dilemma . ${\ displaystyle B> A> D> C}$

Payout dominance

If this happens, there is a risk of miscoordination, since it is no longer possible to distinguish the equilibria with regard to their payouts. But if that is, then the Nash equilibrium is payout-dominant . Rationally acting players 1 and 2 thus choose the strategies and . ${\ displaystyle A = D}$ ${\ displaystyle A <D}$ ${\ displaystyle (x_ {2}, y_ {2})}$ ${\ displaystyle x_ {2}}$ ${\ displaystyle y_ {2}}$

Risk dominance

The concept of risk dominance is used when the solution to a game does not appear clear because there is no clear balance. One tries to solve this problem with the help of different possibilities. One is risk dominance, which examines which equilibrium is least risky (risk dominant ). In the above-mentioned symmetrical bimatrix, the combination of strategies is dominated by risk if: ${\ displaystyle (x_ {2}, y_ {1})}$

{\ displaystyle B + D \ geqslant A + C}

This is the so-called Harsanyi rare criterion ; it is derived from the condition for risk dominance ( ). ${\ displaystyle (BA) ^ {2} \ geqslant (CD) ^ {2}}$

Coordination games

In game theory, a game in which, in contrast to many strategic situations, the focus is not on the conflict, but the actors can achieve the highest payouts by coordinating their behavior, is called a coordination game.

Construction: A coordination game arises with the symmetrical bimatrix mentioned above, if the following applies to player 1:

{\ displaystyle A> B}

and

{\ displaystyle D> C}

and for player 2:

{\ displaystyle a> b}

and

{\ displaystyle d> c}

It follows that (top, left) and (bottom, right) are the two Nash equilibria in pure strategies.

Anti-coordination games

A game is an anti-coordination game if and only if and for player 1 and and for player 2. Because of these restrictions on the payouts (bottom, left) and (top, right) are the two pure Nash equilibria. Also , in order for a switch from (top, left) to (top, right) to increase player 2's payout but decrease player 1's, which creates the conflict. ${\ displaystyle B> A}$ ${\ displaystyle C> D}$ ${\ displaystyle b> a}$ ${\ displaystyle c> d}$ ${\ displaystyle A> C}$