By Michael Bacharach (auth.)
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Since he knows the Japanese payoffs (by element (4)) he could then deduce that he would play f3N· He would then hims~lf play cxN, that is, as maximin dictates. Similarly vice versa. 3 is satisfied. The next step is to see whether what is true in the Bismark Sea game- that maximin strategies are in equilibrium- is true generally. It would be natural to write this as EuA(cxi, f3i)- but cumbersome. We shall retain all the essential information if we simply write uii· For one thing, if we want to know B's payoff it is enough to know that A's is uii• forB's is then immediately given as -uii (the game being ZS).
M is the midpoint of RS, so 0 M = y. M M' must be the arithmetic average of RR' and SS', that is MM' = tRR' +tSS' = tu(y- o)+ tu(y+ o). That is, MM' = Eu(L). 12). In words, the VNM utility of y for sure exceeds MM'. So the curve showing the VNM utilities of sure money receipts passes through a point such as N in the figure, a point above M'. If o is a small quantity, this implies that the marginal utility of sure money is diminishing in the neighbourhood of y. The converse- that diminishing marginal VNM utility of money implies risk aversion- is easily demonstrated by retracing the steps in the above argument.
3 refers to situations in which A has, ex ante, the necessary information to apply the principle to B's problem. 3 if it is applied to the example. S. intelligence believed that the Japanese were about to move a troop and supply convey from the port of Rabaul, on the eastern tip of New Britain, to Lae, to the west, on New Guinea. It might either travel to the north side of New Britain, where visibility would be poor, or to the south, where visibility would be clear. In either case the trip would take three days.