(Silverman’s game) Each of two players chooses a positive integer. If player i’s integer is…

(Silverman’s game) Each of two players chooses a positive integer. 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 strategies, and that the pair of mixed strategies in which each player chooses 1, 2, and 5 each with probability 1/3 is a mixed strategy Nash equilibrium. (In fact, this pair of mixed strategies is the unique mixed strategy Nash equilibrium.)