Decision Making using Game Theory: An introduction for managers

38Games of chanceTable 3.1 Pay-off matrix for a company applying for funding for a new high-speed rail linkSource ofgrant(i)Amount ofgrant (£) a(A)Number ofsuch grants(G)Probabilityof gettingany grant(P)Probability ofgetting this grant(Q = P × G/ΣG i)Expectedvalue (£) a(E = Q × A)EU 4000 3 0.40 0.40 × 3/20 2402000 7 0.40 0.40 × 7/20 2801000 10 0.40 0.40 × 10/20 20020 0.40 720PTRF 2500 10 0.60 0.60 × 5/50 1501500 20 0.60 0.60 × 15/50 270750 30 0.60 0.60 × 30/50 27060 0.50 690a Values are in millions of pounds.insurance would be redundant, since actuaries calculate insurancepremiums so as to make a proWt on average. There must be somethingother than average monetary gain at stake when players play games.For one thing, prudence appears to act as a counterbalance togambling everything on the chance of a huge gain and, furthermore,there may be other than monetary values at stake in the game. Forexample, political pressure may be brought to bear on a company todeter it from accepting the solution with the maximum expectedmonetary pay-oV. Or a company’s creditors (numerous and inXuentialin the case of the Channel Tunnel) may demand the safest option,irrespective of possible gain. Secondly, even in a game where the oddsof winning are slightly in the player’s favour, few people can aVord toplay the game long enough to take advantage of it. Thirdly, the size ofthe outcome may be an important factor and size is a relative thing. InExample 3.2, if the high-speed rail link costs £2 billion, applying to thePublic Transport Reconstruction Fund makes less sense than if the raillink costs £1 billion.Utility theoryExpected utility value is a better principle than expected value when itcomes to guiding players through games of risk. Utility theory assumesthat decisions are made on the basis of what the outcomes are worth to