1 Games in Normal Form (Strategic Form)

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1 Games in Normal Form (Strategic Form) A Game in Normal (strategic) Form consists of three components. A set of players. For each player, a set of strategies (called actions in textbook). The interpretation is that this should be thought of as anything the player can do. 3. For each player, some preferences over the set of strategy pro les. We will let N denote the set of agents, which is assumed to be a nite set. For each player i N we denote by S i the strategy set. To reduce notational clutter we let S = n i=s i (the Cartesian product), which allows us to write a payo function for player i as u i S! R De nition A normal form game is an triple G = (N; S; u) where N = f; ; ng is the set of players, S = n i=s i is the strategy space and u = (u ; ; u n ) are the payo functions for player.. Language/Notation It is convenient to establish some language and shorthand notation for future use. We write s i for a generic element in S i and refer to it as a strategy.. We write s = (s ; ; s n ) for a vector consisting of a strategy for each player. We refer to this as a strategy pro le. 3. We write s i = (s ; ; s i ; s i+ ; ; s n ) for a strategy pro le where the ith coordinate has been removed.. We will also use S i for j6=i S j, so that S i consists of every conceivable s i that the players other than i may pick.

2 5. The latter convention allows us to write (s 0 i; s i ) for the strategy pro le (s ; ; s i ; s 0 i; s i+ ; ; s n ), which will be convenient as equilibrium concepts are de ned by having each player i optimize given s i. Dominant and Dominated Strategies De nition s 0 i is strictly dominated by s 00 i if u i (s 0 i; s i ) < u i (s 00 i ; s i ) for every s i S i. De nition 3 s 0 i is weakly dominated by s 00 i if u i (s 0 i; s i ) u i (s 00 i ; s i ) for every s i S i and if u i (s 0 i; s i ) < u i (s 00 i ; s i ) for some s i S i Remark This is nonstandard in that the typical notion of dominance allows for randomizations. We will return to this when talking about randomized strategies. Remark s 0 i is strictly (weakly) dominated by s 00 i or s 00 i used interchangeably. strictly (weakly) dominates s 0 i are De nition s 0 i is a strictly dominant strategy if u i (s 0 i; s i ) > u i (s i ; s i ) for every s i S i and every s i S i n fs 0 ig De nition 5 s 0 i is a weakly dominant strategy if u i (s 0 i; s i ) u i (s i ; s i ) for every s i S i and every s i S i n fs 0 ig and if for every s i S i n fs 0 ig u i (s 0 i; s i ) > u i (s i ; s i ) for some s i S i... The Prisoners Dilemma Suppose that N = f; g ; S = S = fdefect, Cooperateg and that u (Defect, Cooperate) > u (Cooperate, Cooperate) > u (Defect, Defect) > u (Cooperate, Defect) u (Cooperate, Defect) > u (Cooperate, Cooperate) > u (Defect, Defect) > u (Defect, Cooperate)

3 Let u (Defect, Cooperate) = u (Cooperate, Defect) = a u (Cooperate, Cooperate) = u (Cooperate, Cooperate) = b u (Defect, Defect) = u (Defect, Defect) = c u (Cooperate, Defect) = u (Defect, Cooperate) = d; where a > b > c > d in order to capture the prisoner s dilemma ranking. It is convenient to collect this information in a payo bi-matrix Player Player Defect Cooperate Defect c; c a; d Cooperate d; a b; b and since c > d and a > b it is a strictly dominant to play defect. Hence, (Defect, Defect) is an equilibrium in dominant strategies... An Alternative Story Suppose Alice and Bob have a lawn moving operation. They can either work (hard) or shirk, and the total pro t is 0 if both work hard 6 if one of them works hard and if both shirks. Also assume that the cost of e ort is e and that they split the pro t equally. This gives the following payo matrix Bob Alice Shirk Work hard Shirk ; 3; 3 e Work hard 3 e; 3 5 e; 5 e 3

4 Hence, with cost of e ort e = 3 we get Alice Bob Shirk Work hard Shirk ; 3; 0 Work hard 0; 3 ; ; which is identical a Prisoner s dilemma...3 A Second Price Auction with Bidders (Vickrey Auction) Let N = f; g ; S = S = R + and u (b ; b ) = >< > v b if b > b v b if b = b 0 if b < b u (b ; b ) = >< > v b if b < b v b if b = b 0 if b > b Clearly, there is no strictly dominant strategy as, for example, for every (b 0 ; b 00 ) such that b 0 > b 00 > b u (b 0 ; b ) = u (b 00 ; b ) = v b Proposition The unique weakly dominant strategy is to bid b i = v i Proof. Without loss, consider i = Suppose that v > b then u (v ; b ) = v b > 0 v >< b = u (v ; b ) if b > b u (b ; b ) = v b = u (v ;b ) < u (v ; b ) if b = b > 0 < u (v ; b ) if b < b

5 Suppose that v = b then Suppose that v < b then u (v ; b ) = v b = 0 < v b = 0 if b > v = b u (b ; b ) = 0 if b < v = b u (v ; b ) = v b = 0 v >< b < 0 if b > b u (b ; b ) = v b < 0 if b = b > 0 if b < b Hence, in each case bidding the valuation is optimal, so it is weakly dominant. Next, generalize to the case of N = f; ; ng v i max j6=i b j if b i > max j6=i b j >< v u i (b ; ; b n ) = max j6=i b j if b jarg max jn b jj i = max j6=i b j > 0 if b i < max j6=i b j The proof is identical once we replace b with max j6=i b j, so again it is a weakly dominant strategy to bid the valuation... The Groves Mechanism Suppose that N = f; ; ng ; that x X is some social decision with cost function C (x) and that preferences are given by u i (x; v i ) t where v i is a preference parameter that is also an element of some set V i. To generate a normal form assume that each player i may choose an bv i V i and that given strategy pro le 5

6 bv = (bv ; ; bv n ) a social decision x (bv) arg max xx t i (bv) = C (x (bv)) nx u i (x; bv i ) i= C (x) X u i (x (bv) ; bv j ) Hence, we have a normal form game (N; V; ) where the payo i is given by j6=i i (bv) = u i (x (bv) ; v i ) t i (bv) = u i (x (bv) ; v i ) + X j6=i u i (x (bv) ; bv j ) C (x (bv)) By de nition, x (v i ; bv i ) arg max xx " u i (x; v i ) + X j6=i u i (x; bv j ) C (x) # ; so bv i = v i is a weakly dominant strategy. Moreover, pick any function h i j6=i V J! R and note that if then t i (bv) = C (x (bv)) X u i (x (bv) ; bv j ) + h i (bv i ) j6=i i (bv) = u i (x (bv) ; v i ) + X u i (x (bv) ; bv j ) C (x (bv)) h i (bv i ); {z } j6=i constant so bv i = v i is still a weakly dominant strategy...5 Complete versus Incomplete Information Notice that the normal form that we have speci ed assumes that players have complete information about the payo functions. Usually, the way we think about the Groves mechanism is as a way to truthfully elicit preferences. To make sense of this in a complete information setup we can imagine that a planner is uninformed, while the agents know everything about each other. However, when looking at dominance in this particular class of environment it turns out that the analysis carries over to the case with incomplete information. We may discuss this when we study games of incomplete information. 6

7 ..6 Why it Works Internalizing the Externality Suppose that i wouldn t exist. Then the e cient choice would be max xx X u i (x; bv j ) C (x) S i (v i ) j6=i Hence, the external e ect on the rest of the economy from being added to the economy is X u i (x (bv) ; bv j ) C (x (bv)) S i (v i ) ; j6=i so the interpretation of the Groves transfer is that each agent is being paid the externality it generates on all others (if positive) or needs to pay the externality imposed on all others (if negative)...7 A Vickrey Auction is a Special Case of the Groves Mechanism Let. u i (x; v i ) = x i v i. X = x R n +jx i 0 for each iand P n i= x i 3. C (x) = 0 for each x X. V i = R + Then, x (bv) arg max x i bv i x s.t. x i 0 for each i nx x i = Obviously ties can be broken any way, but one solution is that >< if bv x jarg max i (bv) = jn bv jj i = max j bv j > 0 if bv i < max j bv j i= 7

8 Now t i (bv) = X j6=i >< u j (x (bv) ; bv j ) = > 0 if bv i > max j6=i bv j max jarg max jn bv jj j6=i bv j if bv i = max j6=i bv j max j6=i bv j if bv i < max j6=i bv j Hence, the Groves mechanism boils down to an auction where the winner doesn t pay and the loser is paid the announced valuation of the winner. However, just add h i (bv i ) = max j6=i bv j to the transfer and we get >< max j6=i bv j if bv i > max j6=i bv j t i (bv) = jarg max jn bv jj j6=i bv j if bv i = max j6=i bv j > 0 if bv i < max j6=i bv j We can thus conclude that the second price auction is a special case of a Groves mechanism for an environment where a single indivisible object is to be allocated to an agent... Groves Mechanisms for a Binary Public Good Now, assume instead that. u i (x; v i ) = xv i. X = [0; ] 3. C (x) = xc for each x [0; ]. V i = R + Now x i (bv) arg max x[0;] x or (ignoring the indi erence) < x i (bv) = nx bv i i= c! if P n i= bv i c > 0 0 if P n i= bv i c < 0

9 so that < c t i (bv) = which is often referred to as a pivot mechanism. Pj6=i bv i if P n i= bv i c > 0 0 if P n i= bv i c < 0 ;.3 Iterated Elimination of Strictly Dominated Strategies The reader should be warned that the standard approach to dominance as well as iterations of dominance relies on randomized strategies. We will deal with that later. However, if a strategy is strictly dominated it is not a best response to anything the opponent(s) can do. Rationality would therefore rule out play of strictly dominated strategies. But, assuming that all agents understand the structure of the game, then other agents should understand that strictly dominated strategies will not be played, so one should be able to eliminate these. Hence, we can ask again whether there are additional strategies that are strictly dominated given the smaller set of surviving strategies..3. A Simple Example Consider L C R U ; 0 ; 0; D 0; 3 0; ; 0 and note. R is strictly dominated by C. Hence, player (the row player) should make the inference that R will not be played and consider the reduced game L C U ; 0 ; D 0; 3 0; 9

10 . In the reduced game D is strictly dominated by U, so the game that remains is L C U ; 0 ; Player should then reason that player will predict that he will not play the strictly dominated strategy R and that consequently will not play D. 3. Hence, player should pick C and (U,C) is the unique strategy pro le that survives iterated elimination of strictly dominated strategies..3. A Linear Cournot Duopoly Suppose that inverse demand is P (Q) = max f Q; 0g and that two rms produce a homogenous good at a constant unit cost c = Assuming rms maximize pro ts their payo functions are then (q ; q ) = max fq ( q q ) ; q g (q ; q ) = max fq ( q q ) ; q g which immediately allows us to conclude that any q i > is strictly dominated. Hence, the problem for i simpli es to which has solution q i = any q i > is strictly dominated as then q i ( q i q j ) max q i q i ( q i q j ) q j, which is strictly decreasing in q j We therefore conclude that q j = q i ( q i ) < q i ( q i ) q i q j < 0 where the nal inequality holds because solves the monopoly problem max qq ( q) But then we may conclude that q i < is strictly dominated in the reduced game where quantities 0

11 are picked in 0; as qi < implies that q i ( q i q j ) q j = q i q i q i q i where the last inequality holds because maximizes q i q i + q i q j {z } {z } < 0; In general, assume that after n recursions we are left with [a n ; b n ] Then, after n + recursions we are left with [a n+ ; b n+ ] where a n+ = b n b n+ = a n This can be seen by observing that if q i < bn then, b n b n q i ( q i q j ) q j b n b n = q i ( q i b n ) b n + q i ( q i b n ) b n b n b n < 0 <0 b n q i (b n q j ) {z } {z } 0 <0 as bn maximizes q i ( q i b n ) Moreover, if q i > an then a n a n q i ( q i q j ) q j a n a n a n = q i ( q i a n ) a n + q i (a n q j ) {z } {z } 0 >0 as q i ( q i a n ) a n an maximizes q i ( q i a n ) a n a n < 0 Hence, the set of strategies that survive iterated elimination of strictly dominated strategies is q i [a n ; b n ] for every n But 0 a n+ = a n b n+ = a n = a n = b n

12 so a n! a and b n! b where a = a b = b or a = b = 3. Iterated Elimination of Weakly Dominated Strategies Consider L R U ; 0 0; D 0; 0 0; L is weakly (strictly too) dominated. Once it is eliminated nothing more can be eliminated. Hence, the prediction is (U,R) and (D,R). D is weakly dominated. Once it is eliminated L can be eliminated. Hence (U,R) is all that remains. Hence, the order matters. Nash Equilibrium John Nash suggested that a plausible way to de ne equilibria in games is to postulate. Every player does his or her best against what other players are doing (utility maximization), and;. All players have rational expectations. This leads to the following de nition De nition 6 s S is a (pure strategy) Nash equilibrium if u i s i ; s i ui s i ; s i for each i and every s i in S i

13 . Justi cations As we will see in examples, sometimes the Nash equilibrium concept is a bit problematic as it imposes lots of coordination. However, there are several types of justi cations If the players agree in advance to play (a particular) Nash equilibrium, nobody has an incentive to deviate. Mass action/social norms. If a strategic situation occurs frequently, a Nash equilibrium may emerge as a norm for how to play. Predicting non Nash equilibrium play implies that the game theorist understands the game better than the players being modelled. Evolutionary stability. Close connection between Nash equilibria and evolutionary equilibrium concepts.. The Best Response Correspondance In simple games we can use the de nition of Nash directly and check for equilibria by guess and verify. However, when it gets more complicated we need to be more systematic. De nition 7 Given a normal form game G = (N; S; u) the best response correspondence for player i is a mapping i S i S i de ned by i (s i ) = arg max s i u i (s i ; s i ) = fs i in S i such that u i (s i ; s i ) u i (s 0 i; s i ) for all s 0 i in S i g De nition Given a normal form game G = (N; S; u) the best response correspondence s a mapping S S where for every s S we have that (s) = ( (s i ) ; ; n (s i )) 3

14 Notice that i (s i ) in general may be a set as illustrated in the simple example below, Bob where (Left) = fup, Downg It is pretty clear that Alice Left Right Up 0; 0 ; Down 0; 3; 3 ; Proposition s is a (pure strategy) Nash equilibrium if and only if s (s ) for every player i Proof. Suppose that s is a (pure strategy) Nash equilibrium but that s is not in (s ) Then there at least one player such that s i = i s i implying that there exists s 0 i such that u i s 0 i; s i > ui (s ) Hence, there is a contradiction as this says that s is not a Nash equilibrium. for every player i Next, if s (s ) then s i i s i for each player so that s i arg max s i S i u i s i ; s i ; for each player i which means that s is a Nash equilibrium.

15 .3 Finding Nash Equilibria By Using the Best Response Correspondance.3. The Algoritm Consider the Game Bob Alice x y z a ; ; 0 3; b ; 6; ; 7 c 3; 0 0; 0 6; 5 d 3; 0; 0 5; ; which doesn t have any particular interpretation. Begin by putting in the best responses for Alice by underlining the corresponding payo in the payo bi-matrix as follows Bob x y z Alice a ; ; 0 3; b ; 6; 7; 7 c 3; 0 0; 0 6; 5 d 3; 0; 0 5; ; Do the same for Bob Bob Alice x y z a ; ; 0 3; b ; 6; 7; 7 c 3; 0 0; 0 6; 5 d 3; 0; 0 5; ; We conclude that (c; x) ; (d; x) and (b; x) are Nash equilibria. 5

16 .3. Cournot Duopoly with Linear Demand and Constant Unit Cost Zero Consider the Cournot duopoly model with inverse demand P (q) = q and C i (q i ) = cq i. Then (ignoring (q ; q ) such that q + q >, which cannot be the case in an equilibrium). (q ; q ) = q ( q q ) (q ; q ) = q ( q q ) We will solve for a Nash equilibrium by using the best response. Hence, B (q ) is found by solving In a Nash equilibrium (q ) = arg max q q ( q q ) = q (q ) = arg max q ( q q ) = q q q = (q ) = q q = (q) = q Solving we obtain (q ; q ) = 3 ; 3, which is the unique Nash equilibrium of the game. 6