By Ruey S. Tsay
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Extra info for Analysis of Financial Time Series (Wiley Series in Probability and Statistics - Applied Probability and Statistics Section Series)
For each value of x, determine all saddlepoints of the game, if any. 3. Mixed Strategies Consider the following zero-sum game: T B Â L R Ã 3 2 : 1 4 (a) Show that this game has no saddlepoint. (b) Find a mixed strategy (randomized choice) of (the row) player 1 that makes his expected payoff independent of player 2’s strategy. (c) Find a mixed strategy of player 2 that makes his expected payoff independent of player 1’s strategy. (d) Consider the expected payoffs found under (b) and (c). What do you conclude about how the game could be played if randomized choices are allowed?
3 The numbers in this table are the total payoffs to coalitions and not the net payoffs compared to the coalition members staying alone instead of cooperating. These would be, respectively, 0, 0, 0, 7, 12, 0, and 13. 5 1 Introduction Bargaining Games Bargaining theory focusses on agreements between individual players. 1 A Division Problem Story Consider the following situation. Two players have to agree on the division of one unit of a perfectly divisible good, say a liter of wine. ˛; ˇ/ where ˛; ˇ 0 and ˛Cˇ Ä 1, then they split up the one unit according to this agreement; otherwise, they both receive nothing.
The minimum is reached at the point of intersection of e1 and e4 in the diagram, which has coordinates . 10 ; 118 /. 19 19 118 Hence, the value of the game is 19 , and the unique optimal (minimax) strategy of player 2 is q D . 10 ; 9 /. p1 ; p2 ; p3 ; p4 / of player 1, it follows from the diagram that p2 D p3 D 0, otherwise for q D 10 the value 118 19 19 of the game is not reached, so that p is not a maximin strategy. So we look for a combination of e1 and e4 that gives at least 118 for every q, hence it has to be equal 19 to 118 for every q.
Analysis of Financial Time Series (Wiley Series in Probability and Statistics - Applied Probability and Statistics Section Series) by Ruey S. Tsay