Decision Making using Game Theory: An introduction for managers

133Solving games without Nash equilibrium points using mixed strategiesfocused bias, the pro-mutual lobby would do well at the expense of the other,since community-focus was already the strength of the status quo, although thepro-change lobby would gain some small measure of credibility (1, 4). If bothfactions suggested a criterion-based focus for future business, the pro-mutualfaction would just about prevail (1, 2). However, if the pro-mutual lobby oVereda criterion-based focus, and the pro-change lobby did not, the latter would farebetter, since it would totally undermine the argument for mutuality (3, 0). Andif the pro-mutual group oVered a continuation of community-focused serviceand the pro-change group oVered a change to criterion-based service, the votewould probably go the way of change (2, 1).As things turned out, the action failed and Standard Life remains a mutualsociety to this day, but what strategy should each side have adopted, if they hadaccepted this analysis?The game represented by Figure 6.18 has no Nash equilibrium in purestrategies, but it does in mixed strategies and so can be solved.Let the probabilities p 1 , p 2 , q 1 and q 2 be as indicated, with:p 2 1 p 1 and q 2 1 q 1If Ψ 1 represents the expected pay-oV functions for the prodemutualisation(pro-change) lobby and Ψ 2 represents the expectedpay-oV functions for the pro-mutual lobby, then:Ψ 1 p 1 q 1 1 p 1 q 2 3 p 2 q 1 2 p 2 q 2 1 p 1 q 1 3p 1 (1 q 1 ) 2(1 p 1 ) q 1 (1 p 1 )(1 q 1 )3p 1 q 1 2p 1 q 1 1andΨ 2 p 1 q 1 4 p 1 q 2 0 p 2 q 1 1 p 2 q 2 2 4p 1 q 1 (1 p 1 ) q 1 (1 p 1 )(1 q 1 )2 5p 1 q 1 2p 1 q 1 2So,δΨ 1δp 1 –3q 1 2 0andδΨ 2δq 2 5p 1 1 0