April 6, 2015
Notation for Strategic Form Games Definition A strategic form game (or normal form game) is defined by 1 The set of players i = {1,..., N} 2 The (usually finite) set of actions A i for each player i, a i A i. Then a A = N i=1 A i is called an action profile and it can be written as a = (a i, a i ), where a i A i 3 The preferences over action profiles, or payoff functions u i : A R. In this course, we assume that payoff functions are in the expected utility form. Hence, a strategic form game is Γ = [N, {A i }, {u i ( )}].
Some Examples of Games Example Travelers dilemma: 2 travelers are in the airport and their bags have been lost with. The travelers are asked to claim a value for their lost items but they must say a number between 2 and 100. Each receives compensation equal to the smaller of the two numbers. In addition the traveler claiming the lower value receives a $2 bonus from the other.
Some Examples of Games Example Trading envelopes: There are two envelopes each containing some unspecified amount of money. There are two players and each receives an envelope. The players look into their envelopes and decide whether they want to trade. They exchange envelopes if and only if they both agree to trade. Each player keeps the money in the envelope he leaves with.
Some Examples of Games Example Sequential Bargaining There is a dollar to be divided between two players, provided they can agree on a division. In alternating turns they will propose divisions until one is accepted and then implemented. If they never agree they each receive zero.
Mixed Strategies Definition A mixed strategy for player i is a probability distribution α i over A i. We use the notation α i (a i ) 0 for the probability assigned to action a i. If α i (a i ) = 1 for some action a i then α i is called a pure strategy. A mixed strategy profile is α = (α 1,..., α N ). It can be regarded as a probability distribution over A. Independence. The payoff to player i from a mixed strategy profile α is ) ( N u i (α) = α(a)u i (a) = α i (a i ) a A a A i=1 u i (a)
Independent Mixing Independence makes sense if we interpret mixed strategies as literal randomizations and the players behave non-cooperatively. Sometimes we attach a different interpretation to mixed profiles α i : the conjecture player i holds about the play of his opponents. In that case it is natural to allow correlation. This will come up shortly when we discuss rationalizability.
Solution Concepts Game theory is primarily the study of solution concepts. Formal theories of outcomes Usually based on intuitive ideas Formally a solution concept is a function which takes as inputs the data of the game and produces as an output a mixed strategy profile, possibly a set of them, or possibly something more complicated than that. The first solution concepts we will study are based on the idea of rationality and knowledge of rationality.
Strictly Dominated Strategies A rational player would never play an action which is dominated. Definition An action a i A i is strictly dominated if there is a mixed action α i u i (α i, a i ) > u i (a i, a i ), a i A i. Noteworthy features of the definition We don t quantify over all mixed profiles α i. We do consider potentially dominating mixed actions α i.
Weak Dominance Definition An action a i A i is weakly dominated if a i A i st 1 u i (a i, a i ) u i (a i, a i ), a i A i, 2 u i (a i, a i ) > u i (a i, a i ), for some a i A i.
Examples Prisoner s Dilemma Traveler s Dilemma First-Price Auction Second-Price Auction
Iterative Removal of Dominated Strategies If a rational player never plays dominated strategies, And each player knows the others are rational, Then each player knows that no other player plays dominated strategies, So we need only quantify over the subset of undominated strategies. Etcetera. This defines the solution concept of Iterative Removal of Dominated Strategies. This is a more compelling theory for strictly dominated strategies than for weakly dominated strategies.
Never-Best-Replies A rational player forms a belief and plays a best-reply. Definition An action a i A i is a best-reply (BR) to the profile a i A i if u i (a i, a i ) u i (a i, a i ), a i A i. We say that a i is never a best-reply against pure action profiles if there is no profile a i A i against which a i is best-reply.
Never-Best versus Dominance Proposition If an action a i is dominated, then it is never best. What about the converse?
An Example Example A two player game. L R A 3, 0, B 0, 3, C 1, 1, Figure: Example C is never best. Is it dominated? Not by any pure strategy. However, consider Player 1 s mixed strategy (0.5, 0.5, 0). It earns a higher expected payoff than C against any action of player 2.
Never-Best versus Dominance Does never-best imply dominated by a (possibly) mixed strategy?
Another example Example A two player game (fig 0.2): L R A 3, 10, B 10, 3, C 1, 1, Figure: Example Here C is never a best reply to any action of player 2. But it is not dominated, not even by a mixed strategy. However, note that C is a best-reply to the mixed strategy (1/2, 1/2) of player 2.
Never-Best Replies Revisited Definition An action a i A i is a best-reply (BR) to a mixed-action profile α i if u i (a i, α i ) u i (a i, α i ), a i A i. We say that a i is never a best-reply against mixed actions if there is no mixed action profile α i against which a i is best-reply.
Theorem In a two-player game, a pure action is never best (against mixed actions) iff it is dominated (possibly by a mixed action).
Proof of Theorem We need to show that if a pure action is never best then it is dominated. We prove the contrapositive: if an action is not dominated then it is a best-reply to some mixed action. Let a 1 be a pure action which is not dominated. We want to show that it is a best response to some strategy α 2 A 2. Enumerate the set A 2 st A 2 = {1,... K }. For any α 1 A 1, define the vector of payoffs U α1 = [u 1 (α 1, 1), u 1 (α 1, 2)..., u 1 (α 1, K )] Let V be the collection of these vectors, Note that V is a convex set. V = {U α1 : α 1 A 1 }.
Define the collection of vectors U as follows, U = {U a1 } V = {u R K : u = U a1 v for some v V }. Note that 0 = U a1 U a1 belongs to U. Define the negative orthant of R K as follows. R K = {v R K : v k < 0 for all k = 1..., K } Since a 1 is not dominated, it must be that for every α 1 A 1 there exists some strategy k of player 2 such that u 1 (a 1, k) u 1 (α 1, k) 0. This means that every vector in U has at least one non-negative coordinate, i.e. R K U =.
Since the negative orthant is also a convex set, we can apply the separating hyperplane theorem. Hence, there exists a vector α 2 R K st α 2 = 0 and a real number c such that u α 2 c, u U and v α 2 c, for all v R K. In fact, it must be that c = 0. This is because 0 U so that c 0 and for every ε > 0, the vector v ε = ( ε, ε,... ε) R K and α 2 v ε = ε k (α 2 ) k which can be arbitrarily close to 0 for ε small enough.
Now since v α 2 0 for all v R K, it must be that every coordinate of α 2 is non-negative (otherwise we could find a vector v R K such that v α 2 > 0.) And since α 2 = 0, we can normalize α 2 by ˆα 2 (k) = α k 2 K k=1 αk 2 0 and then K k=1 ˆα 2(k) = 1. Hence the normalized ˆα 2 is a mixed strategy of player 2. Note that the normalization preserves the inequality: u ˆα 2 0, u = U a1 U α1 U. This inequality implies that ˆα 2 U a1 ˆα 2 U α1 for all α 1 A 1. And this is equivalent to u(a 1, ˆα 2 ) u(α 1, ˆα 2 ) for all α 1 A 1. In other words a 1 is a best-reply to ˆα 2.
Three Players Where does the proof break down if there are three or more players? Henceforth: Mixed action profiles α i Π j =i A j (independent) versus Conjectures α i A i (possibly correlated) An action is strictly dominated if and only if there is no conjecture against which it is a best-reply (i.e. never best against possibly correlated conjectures.)
Best-Replies to Conjectures Definition For given α i (possibly correlated) denote the set of best responses in pure strategies: If BR i (α i ) = {a i A i : u i (a i, α i ) u i (a i, α i ), a i }. A i is finite or A i is compact and u i (, α i ) is continuous, then BR i (α i ) is non-empty (and compact).
Rationality and Knowledge of Rationality A rational player forms a conjecture and plays a best-reply. A player who knows that his opponents are rational knows that his opponents play only best-replies to their conjectures. A rational player who knows that his opponents are rational but doesn t know their conjectures forms a refined conjecture and plays a best-reply. etcetera. The conjecture is refined because it rules out strategies for the opponents which are never-best replies.
Rationalizability These ideas lead us to a solution concept based on common knowledge of rationality. Definition A strategy is rationalizable if it survives the iterated removal of never-best responses.
Rationalizability Define inductively Λ 0 i = A i. Then for each k 1, Λ k i = α i Λ k 1 i BR i (α i ) Λi = k Λ k i The set Λi is called the set of rationalizable actions.
A Sequence of Nested Subsets Λ k+1 i Λ k i Trivially Λ 1 i A i = Λ 0 i. Suppose Λ k i Λ k 1 i for all i. Then Λ k i Λ k 1 i, Hence Λ k i Λ k 1 i So that Λ k+1 i = α i Λ k i BR i (α i ) α i Λ k 1 i BR i (α i ) = Λ k i.
Some Basic Properties If A is finite, then the process converges to a non-empty set of rationalizable profiles: since there are finite elements and Λ k+1 i Λ k i, there exists finite k such that Λ k i = Λ k 1 i. If A i is infinite, but compact and if the payoff functions are continuous, then the Λ k i are nested compact sets (requires some work to show.) This implies that the infinite intersection is non-empty.
Fixed Point Definition A product set N i=1 Z i = Z A has the best-reply property if for each i, and for all a i Z i, there is an α i Z i such that a i BR i (α i ). ie Z i α i Z i BR i (α i ). Another way of capturing rationality and knowledge. Suppose it is understood that players play action profiles in Z. This is consistent with rationality only if Z has the best-reply property.
Fixed Point Proposition With finite action spaces (or compact action spaces and continuous payoffs), Λ has the BR-property. Obvious when A is finite. Can be shown by a compactness argument when A i are compact and u i are continuous.
Largest Fixed Point Rationalizability amounts to assuming nothing more than rationality (and common knowledge of rationality.) Proposition Λ is the largest set in A with the BR-property. To prove this note that any set with the best-reply property remains included in Λ k for every k.
Summary 1 Rationalizability captures the implications of rationality and common knowledge of rationality. 2 It is equivalent to iterative removal of strictly dominated strategies. 3 However with three or more players this is true only if conjectures allow for correlation. 4 The rationalizable set is the largest set with the best-reply property. 5 It is thus the most conservative prediction we can make based on assuming only rationality and common knowledge of rationality.
Caution Arguably a player who is rational and who believes that anything is possible should not play a weakly dominated strategy. L R A 1, 3 2, 0 B 1, 1 0, 0
Can we apply something like common knowledge of rationality and caution? L R A 1, 3 2, 0 B 1, 1 0, 0 1 If P1 does not assume anything about 2 s behavior (cautious), then A weakly dominates B. 2 But P1 knows that 2 is rational, she knows that R is not played (since it is strictly dominated by L), thus A and B are equivalent for P1.
Iterative Elimination of Weakly Dominated Strategies (IWD) An attractive concept, but with difficulties. Shaky foundations. Order of elimiation matters. May be too good to be true.
Order of Elimination Matters for IWD Example In the following game we see that order of elimination of WD strategies matters: 1 if we first eliminate R, we get 2 solutions. 2 if we first eliminate B, now R can be eliminated and we have unique solution. L R A 1, 3 2, 0 B 1, 1 0, 0 Figure: Order matters
Too Good to be True Example Take the battle of Zaxes game below. N S N 3, 1 0, 0 S 0, 0 1, 3 Figure: The Battle of Zaxes It does not have any dominated strategies.
Burning Money Now, we modify the game slightly by adding the possibility that Player 1 can publicly burn a small amount of money before they choose actions. It would be surprising if this would matter, but as we see, if ε 1/2 then it gives a unique IWD solution. Now player 1 will have 4 strategies: burn and N (BN), don t burn and N (DN) etc. Also, player 2 will have 4 possible strategies, since strategies must specify her action in 2 possible cases: if 1 burns and if not (ie NS means N if burns, S if not). This way we get the following game. NN NS SN SS BN 3 ε, 1 3 ε, 1 ε, 0 0, 0 DN 3, 1 0, 0 3, 1 0, 0 BS ε, 0 ε, 0 1 ε, 3 1 ε, 3 DS 0, 0 1, 3 0, 0 1, 3 Figure: The Battle of Zaxes with the possibility that player 1 burns money
IWD solution Solving the game by IWD (when ε = 1/2): 1 BS is weakly dominated by a mixed strategy, which gives 0.5 probability to both DS and DN. Eliminate BS. 2 SN and SS are weakly dominated: by NN and NS respectively. Eliminate SN and SS. 3 DS is weakly dominated by BN. Eliminate DS. 4 NS is weakly dominated by NN. Eliminate NS. 5 BN is weakly dominated by DN, which leaves us with unique undominated solution (DN, NN). So, just the possibility of burning money gives player 1 her favorite solution. (She doesn t actually burn the money in that solution.)