Game Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012

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1 Game Theory Lecture Notes By Y. Narahari Departmet of Computer Sciece ad Automatio Idia Istitute of Sciece Bagalore, Idia July 01 Chapter 4: Domiat Strategy Equilibria Note: This is a oly a draft versio, so there could be flaws. If you fid ay errors, please do sed to hari@csa.iisc.eret.i. A more thorough versio would be available soo i this space. I this chapter, we start aalyzig strategic form games by defiig the otio of domiat strategies ad domiat strategy equilibria. There are three otios of domiace that are aptly called strog domiace, weak domiace, ad very weak domiace. First we itroduce strog domiace ad weak domiace ad provide several examples. We itroduce very weak domiace towards the ed of the chapter. 1 Strog Domiace Strogly Domiated Strategy Give a game Γ = N,(S i ),(u i ), a strategy s i S i is said to be strogly domiated if there exists aother strategy s i S i such that u i (s i,s i) > u i (s i,s i ) s i S i I such a case, we say strategy s i strogly domiates strategy s i. Strogly Domiat Strategy A strategy s i S i is said to be a strogly domiat strategy for player i if it strogly domiates every other strategy s i S i. That is, s i s i, Strogly Domiat Strategy Equilibrium u i (s i,s i) > u i (s i,s i ) s i S i A profile of strategies (s 1,s,...,s ) is called a strogly domiat strategy equilibrium of the game Γ = N,(S i ),(u i ) if i = 1,,...,, the strategy s i is a strogly domiatig strategy for player i. 1

2 Example: Prisoer s Dilemma Recall the prisoer s dilemma problem where N = {1,} ad S 1 = S = {C,NC} ad the payoff matrix is give by: 1 NC C NC, 10, 1 C 1, 10 5, 5 Note that the strategy NC is strogly domiated by strategy C for player 1 sice u 1 (C,NC) > u 1 (NC,NC) u 1 (C,C) > u 1 (NC,C) Similarly, the strategy NC is strogly domiated by strategy C for player sice u (NC,C) > u (NC,NC) u (C,C) > u (C,NC) Thus C is a strogly domiat strategy for player 1 ad also for player. Therefore (C,C) is a strogly domiat strategy equilibrium for this game. Note that if a (ratioal) player has a strogly domiatig strategy the we should expect the player to choose that strategy. O the other had, if a player has a strogly domiated strategy, the we should expect the player ot to play it. Weak Domiace Give a game Γ =< N,(S i ),(u i ) >, a strategy s i S i is said to be weakly domiated by a strategy s i S i for player i if for all s i S i, u i (s i,s i ) u i (s i,s i ) s i S i ad u i (s i,s i ) > u i (s i,s i ) for some s i S i Note that strict iequality is satisfied for at least oe s i. The strategy s i strategy s i. is said to weakly domiate Weakly Domiat Strategy A strategy s i is said to be a weakly domiat strategy for player i if it weakly domiates every other strategy s i S i. Weakly Domiat Strategy Equilibrium Give a game Γ = N,(S i ),(u i ), a strategy profile (s 1,...,s ) is called a weakly domiat strategy equilibrium if for i = 1,...,, the strategy s i is a weakly domiat strategy for player i.

3 Example: Modified Prisoer s Dilemma Cosider the followig payoff matrix of a slightly modified versio of the prisoer s dilemma problem. 1 NC C NC, 10, C, 10 5, 5 It is easy to ote that C is a weakly domiat strategy for player 1 ad also for player. Therefore the strategy profile (C,C) is a weakly domiat strategy equilibrium. 3 Examples 3.1 Example: Tragedy of the Commos Recall that N = {1,,...,} is a set of farmers S 1 = S = = S = {0,1} 1 correspods to keepig a sheep, ad 0 correspods to ot keepig a sheep. Keepig a sheep gives a beefit of 1. However, whe a sheep is kept, damage to the eviromet is 5. This damage is equally shared by all the farmers. For i = 1,,..., Case 1: < 5. Give ay s i S i, sice < 5, ( 5 u i (s 1,...,s ) = s i 5 ( ) 5 = s i 5 j=1 u i (0,s i ) = 5 ( ) 5 u i (1,s i ) = 5 ) < 0, ad therefore, ui (0,s i ) > u i (1,s i ) s i S i. This implies that B i (s i ) = {0} i N This meas (0,0,...,0) is a strogly domiat strategy equilibrium. That is, there is o icetive for ay farmer to keep a sheep. Case : = 5. Here u i (0,s i ) = 5 u i (1,s i ) = 5 3

4 Thus u i (0,s i ) = u i (1,s i ), s i S i It is easy to see that oe of the strategies here is a weakly domiat strategy or a strogly domiat strategy. Case 3: > 5. Here u 1 (0,s i ) = 5 u i (1,s i ) = 5 5 Thus u i (1,s i ) > u i (0,s i ) s i S i Hece (1,1,...,1) is a strogly domiat strategy equilibrium. Thus if > 5, it is good for all the farmers to keep a sheep. Now if the Govermet decides to impose a pollutio tax of 5 uits for each sheep kept, we have u i (s 1,...,s ) = s i 5s i 5 = 4s i 5 s i 5 j=1 Here u i (0,s i ) = 5 u i (1,s i ) = This meas whatever the value of, (0,0,...,0) is a strogly domiat strategy equilibrium. This is bad ews for the farmers. 3. Example: Braess Paradox Game Recall the Braess paradox game with additioal capacity itroduced from A to B. I this game, it ca be show, for every player i, that u i (AB,s i ) > u i (A,s i ) s i S i u i (AB,s i ) > u i (B,s i ) s i S i This shows that (AB,AB,...,AB) is a strogly domiat strategy equilibrium. Note that the above equilibrium profile leads to a total delay of 40 miutes. O the other had, if 500 vehicles use the strategy A ad ad the other 500 vehicles use the strategy B, the total delay for each vehicle is oly 35 miutes. The paradox here is that the itroductio of a additioal lik forces the strategy of AB o every vehicle (AB beig a strogly domiat strategy for each vehicle) thereby leadig to a delay that is higher tha what it would be for a o-equilibrium profile. 4

5 3.3 Example: Secod Price Sealed Bid Auctio with Complete Iformatio Cosider the secod price sealed bid auctio for sellig a sigle idivisible item discussed Example... Let b 1,b,...,b be the bids (strategies) ad we shall deote a bid profile (strategy profile) by b = (b 1,b,...,b ). Assume that v i,b i (0, ) for i = 1,,...,. Recall that the item is awarded to the bidder who has the lowest idex amog all the highest bidders. Recall the allocatio fuctio: The payoff for each bidder is give by: y i (b 1,...,b ) = 1 if b i > b j for j = 1,,...,i 1 ad = 0 else. b i b j for j = i + 1,..., u i (b 1,...,b ) = y i (b 1,...,b )(v i t i (b 1,...,b )) where t i (b 1,...,b ) is the amout paid by the wiig bidder. Beig secod price auctio, the wier pays oly the ext highest bid. We ow show that the strategy profile (b 1,...,b ) = (v 1,...,v ) is a weakly domiat strategy equilibrium for this game. Proof: Cosider bidder 1. His value is v 1 ad bid is b 1. The other bidders have bids b,...,b ad valuatios v,...,v. We cosider the followig cases. Case 1: v 1 max(b,...,b ). There are two sub-cases here: b 1 max(b,...,b ) ad b 1 < max(b,...,b ). Case : v 1 < max(b,...,b ). There are two sub-cases here: b 1 max(b,...,b ) ad b 1 < max(b,...,b ). We aalyze these cases separately below. Case 1: v 1 max(b,...,b ). We look at the followig scearios. Let b 1 max(b,...,b ). This implies that bidder 1 is the wier, which implies that u 1 = v 1 max(b,...,b ) 0. Let b 1 < max(b,...,b ). This meas that bidder 1 is ot the wier, which i tur meas u 1 = 0. Let b 1 = v 1, the sice v 1 max(b,...,b ), we have u 1 = v 1 max(b,...,b ). Therefore, if b 1 = v 1, the utility u 1 is greater tha or equal to the maximum utility obtaiable. Thus, whatever the values of b,...,b, it is a best respose for player 1 to bid v 1. Thus b 1 = v 1 is a weakly domiat strategy for a bidder 1. 5

6 Case : v 1 < max(b,...,b ). As before, we look at the followig scearios. Let b 1 max(b,...,b ). This implies that bidder 1 is the wier ad the payoff is give by: u 1 = v 1 max(b,...,b ) < 0. Let b 1 < max(b,...,b ). This meas bidder 1 is ot the wier. Therefore u 1 = 0. If b 1 = v 1, the bidder 1 is ot the wier ad therefore u 1 = 0. From the above aalysis, it is clear that b 1 = v 1 is a best respose strategy for player 1 i Case also. Combiig our aalysis of Case 1 ad Case, we have that u 1 (v 1,b,...,b ) u 1 ( ˆb 1,b,...,b ) ˆb 1 S 1 b S,..., b S Also, we ca show (ad this is left as a exercise) that, for ay b 1 v 1, we ca always fid b S,b 3 S 3,...,b S, such that u 1 (v 1,b,... b ) > u 1 (b 1,b,...,b ). Thus b 1 = v 1 is a weakly domiat strategy for a bidder 1. Usig almost similar argumets, we ca show that b i = v i is a weakly domiat strategy for bidder i where i =,3,...,. Therefore (v 1,...,v ) is a weakly domiat strategy equilibrium. 4 Very Weak Domiace Give a game Γ =< N,(S i ),(u i ) >, a strategy s i S i is said to be very weakly domiated by a strategy s i S i for player i if for all s i S i, u i (s i,s i) u i (s i,s i ) s i S i Note that strict iequality eed ot be satisfied for ay s i as i the case of weak domiace. The strategy s i is said to very weakly domiate strategy s i. Very Weakly Domiat Strategy A strategy s i is said to be a very weakly domiat strategy for player i if it weakly domiates every other strategy s i S i. Example: Modified Prisoer s Dilemma - Versio Cosider the followig payoff matrix of aother modified versio of the prisoer s dilemma problem. 1 NC C NC, 5, C, 10 5, 5 It is easy to ote that C is a very weakly domiat strategy for player 1 while C is a weakly domiat strategy for player. The strategy profile (C,C) ow cosists of a very weakly domiat strategy (for player 1) ad a weakly domiat strategy (for player ). We ofte use the otio of very weak domiace i mechaism desig settigs (Part of the book). 6

7 5 To Probe Further The material discussed i this chapter is maily take from the books by Myerso [1]; Mascolell, Whisto, ad Gree []; Shoham ad Leyto-Brow [3]. 6 Problems 1. Cosider the followig istace of the prisoers dilemma problem. Fid the values of x for which: 1 NC C NC 4, 4, x C x, x, x (a) the profile (C,C) is a strogly domiat strategy equilibrium. (b) the profile (C,C) is a weakly domiat strategy equilibrium but ot a strogly domiat strategy equilibrium. (c) the profile (C,C) is a ot eve a weakly domiat strategy equilibrium. I each case, say whether it is possible to fid such a x. Justify your aswer i each case.. First Price Auctio. Assume two bidders with valuatios v 1 ad v for a object. Their bids are i multiples of some uit (that is, discrete). The bidde with higher bid wis the auctio ad pays the amout that he has bid. If both bid the same amout, oe of them gets the object with equal probability 1. I this game, (a) Are ay strategies strogly domiated? (b) Are ay strategies weakly domiated? 3. There are departmets i I.I.Sc. Each departmet ca try to covice the Director to get a certai budget. If h i is the umber of hours of work put i by a departmet to make the proposal ad c i = w i h i is cost of this effort to the departmet, where w i is a costat. Whe the effort levels of the departmets are (h 1,h,...,h ), the total budget that gets allocated to all the departmets is: α h i + β i=1 where α ad β are costats. Cosider a game where the departmets simultaeously ad idepedetly decide how may hours to sped o this effort. Show that a strictly domiat strategy equilibrium exists iff β = 0. Compute this equilibrium. 4. Complete the proof that reportig true values i Vickrey auctio is a weakly domiat strategy equilibrium. 5. I Case ( = 5) of the tragedy of the commos game, ivestigate whether ay of the strategies is a very weakly domiat strategy. i=1 h i 7

8 6. Compute strogly or weakly domiat strategy equilibria of the Braess paradox game whe the umber 5 is replaced by the umber 0. Refereces [1] Roger B. Myerso. Game Theory: Aalysis of Coflict. Harvard Uiversity Press, Cambridge, Massachusetts, USA, [] Adreu Mas-Colell, Michael D. Whisto, ad Jerry R. Gree. Micorecoomic Theory. Oxford Uiversity Press, [3] Yoam Shoham ad Kevi Leyto-Brow. Multiaget systems: Algorithmic, Game-Theoretic, ad Logical Foudatios. Cambridge Uiversity Press, New York, USA, 009,

Notes on Expected Revenue from Auctions

Notes on Expected Revenue from Auctions Notes o Epected Reveue from Auctios Professor Bergstrom These otes spell out some of the mathematical details about first ad secod price sealed bid auctios that were discussed i Thursday s lecture You