Decision Making using Game Theory: An introduction for managers

103Mixed-motive games without single equilibrium points: archetype 1 – leadership gamesStrategyNominee 2DeclinenominationAcceptnominationNominee1DeclinenominationAcceptnomination2, 2 3, 44, 3 1, 1Figure 6.5Pay-off matrix for leadership games.It can be seen from the matrix that there are no dominant or inadmissiblestrategies. Neither candidate can select a strategy that will yield thebest pay-oV no matter what the other candidate does. The minimaxprinciple fails too because, according to it, both candidates shouldchoose their Wrst strategy (decline the nomination) so as to avoid theworst pay-oV (1, 1). Yet, if they do this, both candidates regret it oncethe other’s choice becomes known. Hence, the minimax strategies arenot in equilibrium and the ‘solution’ (2, 2) is not an equilibrium point.It is unstable and both players are tempted to deviate from it, althoughit should be pointed out that the worst case scenario (1, 1) arises whenboth players deviate from their minimax strategies.Despite the failure of both the elimination and the minimax approaches,there are two equilibrium points on the Figure 6.5 matrix. Ifnominee 1 chooses to accept the nomination, nominee 2 can do nobetter than decline; and if nominee 1 chooses to decline the nomination,nominee 2 can do no better than accept. So there are twoequilibrium points – those with pay-oVs (4, 3) and (3, 4).Unlike zero-sum games, the value of the game is not a constantbecause the players do not agree about preferability and the twoequilibrium points are therefore asymmetrical. There is no formalsolution beyond this. Informal factors such as explicit negotiation andcultural prominence must be explored if a more deWnite outcomeis required. For example, a younger nominee may defer in favour ofan older one in companies where seniority is the prominent basisfor promotion; or the two candidates may negotiate a politicalarrangement. Either way, it is in the interests of both players in a