EXERCISE 118.1 (Silverman's game) Each of two players chooses a positive inte- ger. If player i's integer is greater than player j's integer and less than three times this integer, then player j pays $1 to player i. If player i's integer is at least three times player j's integer, then player i pays $1 to player j. If the integers are equal, no payment is made. Each player's preferences are represented by her expected monetary payoff. Show that the game has no Nash equilibrium in pure strate- gies and that the pair of mixed strategies in which each player chooses 1, 2, and 5 each with probability = is a mixed strategy Nash equilibrium. (In fact, this pair of mixed strategies is the unique mixed strategy Nash equilibrium.) (You cannot ap- peal to Proposition 116.2 because the number of actions of each player is not finite. However, you can use the argument for the "if" part of this result.)