Class Note for ECON 431 with Professor Dufwenberg at UA
Class Note for ECON 431 with Professor Dufwenberg at UA
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Date Created: 02/06/15
University of Arizona Department of Economics ECON 431 Spring 2009 Martin DufWenberg Weak Dominance Watson discusses how a rational player will never choose a strictly dominated strategy At times ruling out strictly dominated strategies and even performing iterated elimination of strictly dominated strategies IESDS or focusing on Nash equilibria may have less cutting power than one may deem plausible Consider the following example a b a 11 00 b 00 00 No strategy is strictly dominated all strategies survive IESDS and a a and b b are the game s two Nash equilibria verify Yet choosing b may seem unreasonable as choosing a is never worse and sometimes better The follow de nition picks that up Definition1 Strategy sieSi is weakly dominated if there is a mixed strategy oieASi such that ui6i si Z uisi si for all sie Si and ui6i si gt uisi si for some sie Si Verify that strategy b is weakly dominated in the previous example for either player Note also that ruling out weakly dominated strategies may rule out some Nash equilibria like b b above Let s practice deriving related insights some more Exercise 1 Find all strategies surviving IESDS all Nash equilibria and all strategies which are not weakly dominated in the following game d e a 51 01 b 50 11 c 08 07 The solution concept of iterated elimination of weakly dominated strategies IEWDS can be defined analogously with IESDS except that at each step of the procedure one requires eliminated strategies to be weakly dominated rather than strictly dominated Exercise 2 Apply IEWDS to the above two games 1 Watson brie y discusses weak dominance on page 324 of his book Beware he uses a slightly different definition that I do he does not consider domination by a mixed strategy My definition admits that in order to stay comparable with Watson39s definition of strict domination on page 52 in his book University of Arizona Department of Economics ECON 431 Spring 2009 Martin DufWenberg It turns out IEWDS sometimes has remarkable cutting power Here is an example presented as an exercise which illustrates Exercise 3 Ann and Bob play a game in which each of them simultaneously chooses an integer from the set 0 l 10 Whoever chooses the lowest number wins as many dollars as the number chosen If Ann and Bob choose the same number they share the corresponding number of dollars Model this as a normal form game What strategies survive IESDS What about IEWDS And what about a procedure which does one round of elimination of weakly dominated strategies followed by IESDS this is sometime called the DekelFudenberg procedure Find all Nash equilibria hint there are three Another example where IEWDS has remarkable cutting power is the Battleofthesexes withanoutsideoption discussed in class Here is a related example Exercise 4 Consider the following coordination game x y X 33 00 Y 00 11 Find all Nash equilibria Find all strategies surviving IESDS and IEWDS Then consider instead the following game which adds a kind of outside option for player 1 x y X 33 00 Y 00 11 Z 22 22 Find all Nash equilibria Find all strategies surviving IESDS and IEWDS Isn t it intriguing how according to IEWDS an outside option can be added not used and still affect the outcome relative the case without the outside option The next exercise spins related themes even further Exercise 5 Burning Money game Consider the following battleof thesexes game B F B 31 00 F 00 13 This is a normal form game Draw an extensive form game which has that normal form Then consider a somewhat different extensive form game It is just like the game you just University of Arizona Department of Economics ECON 431 Spring 2009 Martin DufWenberg drew except that before each individual makes a choice of B or F the individual to be modeled as player 1 makes a choice whether or not to throw away one unit of payoff The other individual observes this choice and then the two of them simultaneously choose between B and F The subsequent payoffs are as before except that if player I threw away a payoff unit then his overall payoff is modified accordingly everywhere Model this new situation as a new extensive form game Then write down its normal form Apply IESDS to the game Then apply IEWDS to the game Compare the solutions The notion of forward induction is sometimes used to re ect instances where a player s early choice in a game tree offers hints as to what he may be up to in a later part of a game The IEWDS solution is formulated for a normal form game but once we have the strategies surviving IEWDS in hand we can discuss what behavior they correspond to in the extensive form game from which the normal form was derived Do this for the game in Exercise 4 and give the IEWDS outcome a forward induction interpretation While IEWDS is a fascinating solution concept it is also controversial The following example illustrates one reason d e a 21 01 b 03 50 c 00 04 What strategies survive IESDS Verify that only a and d survive IEWDS Ask yourself is it really reasonable that we have concluded if we apply IEWDS that strategy e will not be used After all e is neither weakly dominated in the overall game nor worse than d given that player 1 uses a strategy surviving IEWDS In other words IEWDS results in the following maximally reduced 1X1 game d a 21 If at this point we bring back strategy e we get the following 1X2 game in which e is not weakly dominated Exercise 6 Show that IESDS when applied to any finite game does not exhibit an analogous problem