Strict balance

In game theory , given a game in normal form, strict equilibrium is a pair of strategies that meet the requirement that both strategies are strictly the best answers to each other. According to the strict equilibrium, each player loses if he is the only one deviating from his equilibrium strategy. If a Nash equilibrium consists only of dominant strategies, it is also referred to as a strict equilibrium. This means that the player cannot improve his payout by deviating from one side . A strict Nash equilibrium is characterized by the fact that no player can improve himself by changing his strategy unilaterally. The concept of the strict Nash equilibrium comes from John Harsanyi . ${\displaystyle i}$

definition

Designate the strategy space of player 1 and the strategy space of player 2 and the payout function of player 1 and the payout function of player 2, then there is a strict equilibrium if: ${\displaystyle S_{1}}$ ${\displaystyle S_{2}}$ ${\displaystyle u_{1}}$ ${\displaystyle u_{2}}$ ${\displaystyle (s,t)}$

${\displaystyle {\begin{aligned}u_{1}(s,t)>u_{1}(s',t)&\quad \forall \ s'\in S_{1},s\neq s'\\u_{2}(t,s)>u_{2}(t',s)&\quad \forall \ t'\in S_{2},t\neq t'\end{aligned}}}$.

Strict Nash equilibrium

A characteristic of strict equilibrium is that each player loses if he is the only one to deviate from his equilibrium strategy (see Nash equilibrium ). It denotes the set of strategies ( alternative courses of action) of the -th player and the Cartesian product of these sets of strategies. ${\displaystyle \Sigma _{i}}$ ${\displaystyle i}$ ${\displaystyle \Sigma :=\Sigma _{1}\times \dotsb \times \Sigma _{n}}$

Under a Nash equilibrium in pure strategies is defined as a strategy profile in which each player's strategy is a best response is to the selected strategies of the other players. If all other players adhere to their chosen strategies, the Nash equilibrium in pure strategies is formally characterized by the fact that there is no one that promises the player a higher payout: ${\displaystyle \sigma ^{*}=(\sigma _{1}^{*},\dotsc ,\sigma _{n}^{*})\in \Sigma }$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle \sigma _{i}\neq \sigma _{i}^{*}}$ ${\displaystyle i}$

${\displaystyle \forall \sigma _{i}\in \Sigma _{i}:u_{i}(\sigma _{1}^{*},\dotsc ,\sigma _{i}^{*},\dotsc ,\sigma _{n}^{*})>u_{i}(\sigma _{1}^{*},\dotsc ,\sigma _{i},\dotsc ,\sigma _{n}^{*})}$.

Strict balance in evolutionarily stable strategies

The strict equilibrium is a constituent characteristic of an equilibrium in evolutionarily stable strategies (see evolutionarily stable strategy ). If there is a strict equilibrium, it is evolutionarily stable.

Individual evidence

1. ^ Werner Güth: Game theory and economic (accessory) games . 2nd Edition. Springer, Berlin 1999, ISBN 978-3-540-65211-3 , pp. 71 ( limited preview in Google Book search).

See also

• Equilibrium (game theory)