Set 4 quiz of zero-sum games
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1) The document contains solutions to 5 quiz games involving zero-sum games. 2) For game 1, the payoff matrix represents a zero-sum game without an equilibrium point if no cell satisfies both players' best strategies. 3) For game 2, the best strategy for the convict is to choose "forest" with probability n/(n+1) to maximize expected value of escaping. It is a two-person zero-sum game without an equilibrium point.
Set 4 quiz of zero-sum games
- 1. SET 4: QUIZ OF ZERO-SUM GAMES Erwin Widodo
- 2. GAME 1 When do we state that above payoff matrix is a representation of a zero-sum game without equilibrium point? Please elaborate your reason. Player II One Two Player I One a b Two d c
- 3. GAME 2 What is the best strategy if you were the convict in this case? Give the reasons. What is the type of this game? Sheriff Highway Forest Convic t Highw ay 0 1 Forest 1 1-1/n
- 4. GAME 3 Players I and II simultaneously call out one of the numbers: “one” or “two”. Player I wins if the sum of the numbers is odd. Player II wins if the sum of the numbers is even. The amount paid to the winner by the loser is always the sum of the numbers in dollars. The payoff matrix is as follows.
- 5. GAME 3 Problems: Is there any equilibrium point? Give your reason. Suppose Player I play to call “one” 60% and to call “two” 40% of the time, what is the average value? Is that the minimax value? If it isn’t please give your suggested value. Player II’s call One Two Player I’s call One -2 3 Two 3 -4
- 6. GAME 4 Player 1 maximize its payoff, while Player 2 is the opposite. Unfortunately some payoffs are missing. Does this game have its equilibrium point (EP)? Explain your reasoning to determine this EP. Player 2 I II III IV Player 1 A 2 ? ? ? B 3 ? ? ? C 4 ? ? ? D 5 6 7 8
- 7. GAME 5 From a set of 3 cards, numbered as 1, 2, 3, Player 1 select a card at will. Player 2 tries to guess this selected card. After each guess, Player 1 signals High, Low, of Correct depending on Player 2’s guess. The game is over when the card is successfully guessed by Player 2. Player 2 pays Player 1 an amount equals to the number of trials Player 2 made. Problems: What payoff matrix should be constructed? What type of game is it? Please give your reason
- 8. SOLUTION 1 Assume there is no EP and let’s start from p11. If a ≥ b, then b < c, as otherwise b is an EP. Since b < c, we must have c > d, as otherwise c is an EP. Continuing thus, we see that d < a. In other words, if a ≥ b, then a > b < c > d < a. By symmetry, if a ≤ b, then a < b > c < d> a. 20 points
- 9. SOLUTION 2 As the convict, let’s say choose “highway” at p (p)(0)+(1-p)(1)=(p)(1)+(1-p)(1-1/n) p =1/(n+1) and choose “forest” at (1-p)= n/(n+1) The expected value to escape= n/(n+1) Two-person zero-sum game without equilibrium point 15 points 5 points
- 10. SOLUTION 3 Maximin=-2; minimax=3 no EP P1 calls “one” 60% & calls “two” 40%: If P2 calls “one”: 60%(-2)+40%(3)=0 If P2 calls “two”: 60%(3)+40%(-4)=0.2 Hence the “at least value” of P1 in this case = 0 VN Minimax value P1 calls “one” at “p” p(-2)+(1-p)(3)=p(3)+(1-p)(-4) p=7/12 VN Minimax=1/12 8 points 8 points 4 points
- 11. SOLUTION 4 Player 1 viewpoint: no matter “?” In each row of ? > p11,p21,p31,p41 player 1 will choose p11,p21,p31,p41 as minima and p41 as maximin value Player 2 viewpoint: In each row of ? < p11,p21,p31,p41 player 2 will choose p41 and may choose ?s or p42,p43,p44 as maxima but for sure will choose p41 as minimax value Hence the EP is p41 2 points 9 points 9 points
- 12. SOLUTION 5 Maximin=1;Minimax=2Two-person zero-sum game without equilibrium point Player 2 123 132 21/2 3 312 321 Player 1 1 1 1 2 2 3 2 2 3 1 3 2 3 3 2 2 1 1 The payoffs represent the amount Player 1 gets 15 points 5 points