Decision Making using Game Theory: An introduction for managers

44Games of chanceSo the account manager should invest (approximately) 34% in theshare portfolio and 66% in Wxed deposits.The second derivative veriWes that U '(x) 0 represents a localmaximum since:U "(x) 1849(1067 43m) 3/2 1131(1067 29m) 3/2and, clearly, if the case of the negative root of expected value is ignored,U "(x) 0Of course, any relationship between value and utility only makessense if the pay-oV is numerical – which usually means monetary.Although, in theory, it is possible to assign a utility value to any gamebeing played, there is no reason to assume that a relationship of anykind deWnitely exists – linear, logarithmic, root, power or anything else.What is needed is an interval scale (a scale in which the units ofmeasurement and the Wxed points are arbitrarily, but proportionallyspaced) for solving games of risk that do not have numerical outcomes.One theory, proposed by von Neumann and Morgenstern in 1944, isbased on the assumption that a player can express a preference not onlybetween outcomes, but also between any outcome and any lotteryinvolving another pair of outcomes. The upshot of this theory is that itis possible to convert a player’s order of preference among outcomesinto numerical utility values. This is what makes gambling and insuranceboth rational games; the utility value of the gamble, involving theprobable loss of a small stake for the unlikely gain of a large prize, maybe positive, even if the average outcome (the cash value) over a longperiod of time is negative. The von Neumann–Morgenstern utilitytheory thus assigns arbitrary utility values to each player’s least andmost preferred outcomes, like the Wxed points on a temperature scale.The utility values of all the outcomes in between can then be determinedas follows. If the player does not diVerentiate between two ormore outcomes, they are assigned the same utility value. Otherwise,utility values can be assigned to each outcome by comparing eachoutcome, like a yardstick, to a lottery involving the most and leastpreferred. If the player does not distinguish between a lottery with aknown probability and an outcome, then that outcome can be assigned