# lecture17 - Lecture XVII Dynamic Games with Incomplete...

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Unformatted text preview: Lecture XVII: Dynamic Games with Incomplete Information Markus M. M¨obius May 6, 2004 • Gibbons, sections 4.1 and 4.2 • Osborne, chapter 10 1 Introduction In the last two lectures I introduced the idea of incomplete information. We analyzed some important simultaneous move games such as sealed bid auctions and public goods. In practice, almost all of the interesting models with incomplete informa- tion are dynamic games also. Before we talk about these games we’ll need a new solution concept called Perfect Bayesian Equilibrium. Intuitively, PBE is to extensive form games with incomplete games what SPE is to extensive form games with complete information. The concept we did last time, BNE is a simply the familiar Nash equilibrium under the Harsanyi representation of incomplete information. In principle, we could use the Harsanyi representation and SPE in dynamic games of incomplete information. However, dynamic games with incomplete information typi- cally don’t have enough subgames to do SPE. Therefore, many ’non-credible’ threats are possible again and we get too many unreasonable SPE’s. PBE allows subgame reasoning at information sets which are not single nodes whereas SPE only applies at single node information sets of players (because only those can be part of a proper subgame). The following example illustrates some problems with SPE. 1 1.1 Example I - SPE Our first example has no incomplete information at all. 1 L R 2 A B 2 2 1 1 3 3 Its unique SPE is (R,B). The next game looks formally the same - however, SPE is the same as NE because the game has no proper subgames. 1 L R RR 2 A B 2 A B 2 2 1 1 3 3 1 1 3 3 The old SPE survives - all ( pR + (1- p ) RR,B ) for all p is SPE. But there are suddenly strange SPE such as ( L,qA + (1- q ) B ) for q ≥ 1 2 . Player 2’s 2 strategy looks like an non-credible threat again - but out notion of SPE can’t rule it out! Remember: SPE can fail to rule out actions which are not optimal given any ’beliefs’ about uncertainty. Remark 1 This problem becomes severe with incomplete information: moves of Nature are not observed by one or both players. Hence the resulting exten- sive form game will have no or few subgames. This and the above example illustrate the need to replace the concept of a ’subgame’ with the concept of a ’continuation game’. 1.2 Example II: Spence’s Job-Market Signalling The most famous example of dynamic game with incomplete information is Spence’s signalling game. There are two players - a firm and a worker. The worker has some private information about his ability and has the option of acquiring some education. Education is always costly, but less so for more able workers. However, education does not improve the worker’s productivity at all! In Spence’s model education merely serves as a signal to firms. His model allows equilibria where able workers will acquire educa- tion and less able workers won’t. Hence firms will pay high wages only to those who acquired education - however, they do this because education has...
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lecture17 - Lecture XVII Dynamic Games with Incomplete...

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