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Intro to Game Theory

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Intro to Game Theory P n testing some i = 1 f (i) equation Game Theory 15-451/651 10/12/15 Plan for Today - Zero-sum games - Minimax theorem • 2-Player Zero-Sum Games (matrix games) - Connection to maxflow-mincut – Minimax optimal strategies - General-sum games – Connection to randomized algorithms – Minimax theorem – Connection to max-flow / min-cut Game Theory • General-Sum Games (bimatrix games) (and its connections to Algorithm – notion of Nash Equilibrium Analysis and Computer Science) • Proof of existence of Nash Equilibria – using Brouwer’s fixed-point theorem Game theory Setting • Field developed by economists to study social & economic interactions. • Have a collection of participants, or players. – Wanted to understand why people behave the way they • Each has a set of choices, or strategies for do in different economic situations. Effects of incentives. Rational explanation of behavior. how to play/behave. • Combined behavior results in payoffs • “Game” = interaction between parties with their own interests. Could be called “interaction theory”. (satisfaction level) for each player. • Big in CS for understanding large systems: – Internet routing, social networks – Problems like spam etc. All my examples will involve just 2 players (which will make them easier to picture, as will become clear in • And for thinking about algorithms! a moment…) Consider the following scenario… • Shooter has a penalty shot. Can choose to 2-player zero-sum shoot left or shoot right. games • Goalie can choose to dive left or dive right. (aka matrix games) • If goalie guesses correctly, (s)he saves the day. If not, it’s a goooooaaaaall! • Vice-versa for shooter. 1 2-Player Zero-Sum games Game Theory terminolgy • Two players R and C. Zero-sum means that what’s • Rows and columns are called pure strategies. good for one is bad for the other. • Game defined by matrix with a row for each of R’s • Randomized algs called mixed strategies. options and a column for each of C’s options. Matrix tells who wins how much. • “Zero sum” means that game is purely • an entry (x,y) means: x = payoff to row player, y = payoff to competitive. (x,y) satisfies x+y=0. (Game column player. “Zero sum” means that y = -x. doesn’t have to be fair). • E.g., penalty shot: Left Right goalie Left Right goalie Left (0,0) (1,-1) Left (0,0) (1,-1) shooter Goooaaaal! shooter Goooaaaal! Right (1,-1) (0,0) No goal Right (1,-1) (0,0) No goal Game Theory terminolgy Minimax-optimal strategies • Usually describe in terms of 2 matrices R and C, • Minimax optimal strategy is a (randomized) where for zero-sum games we have C = -R. strategy that has the best guarantee on its (I am putting them into a single matrix where each entry is expected gain, over choices of the opponent. a pair, because it is easier visually). [maximizes the minimum] • I.e., the thing to play if your opponent knows you well. Left Right goalie Left Right goalie Left (0,0) (1,-1) Left (0,0) (1,-1) shooter GOAALLL!!! shooter GOAALLL!!! Right (1,-1) (0,0) No goal Right (1,-1) (0,0) No goal Minimax-optimal strategies Minimax-optimal strategies • What are the minimax optimal strategies for • How about penalty shot with goalie who’s this game? weaker on the left? Minimax optimal strategy for both players is Minimax optimal for shooter is (2/3,1/3). Guarantees expected gain at least 2/3. 50/50. Gives expected gain of ½ for shooter Minimax optimal for goalie is also (2/3,1/3). (-½ for goalie). Any other is worse. Guarantees expected loss at most 2/3. Left Right goalie Left Right goalie Left (0,0) (1,-1) Left (½,-½) (1,-1) shooter GOAALLL!!! shooter GOAALLL!!! Right (1,-1) (0,0) No goal Right (1,-1) (0,0) 50/50 2 Minimax-optimal strategies Minimax-optimal strategies • How about if shooter is less accurate on the • In small games we can solve by considering a few left too? cases. • Later, we will see how to solve for minimax optimal Minimax optimal for shooter is (4/5,1/5). in NxN games using Linear Programming. Guarantees expected gain at least 3/5. Minimax optimal for goalie is (3/5,2/5). – poly time in size of matrix if use poly-time LP alg. Guarantees expected loss at most 3/5. Left Right goalie Left (½,-½) (¾,-¾) shooter Right (1,-1) (0,0) Minimax Theorem (von Neumann 1928) Matrix games and Algorithms Adversary • Every 2-player zero-sum game has a unique • Gives a useful way of thinking Alg player about randomized algorithms. value V. • Minimax optimal strategy for R guarantees • Think of rows as different algorithms, columns R’s expected gain at least V. as different possible inputs. • Minimax optimal strategy for C guarantees • M(i,j) = cost of algorithm i on input j. C’s expected loss at most V. Counterintuitive: Means it doesn’t hurt to publish • Algorithm designer goal: good strategy for row your strategy if both players are optimal. (Borel had player. Lower bound: good strategy for adversary. proved for symmetric 5x5 but thought was false for larger One way to think of upper-bounds/lower-bounds: on games) value of this game We will see one proof in a bit… (matrix may be HUGE, but useful conceptually) E.g., hashing Adversary Matrix games and Algs Adversary Alg player Alg player •Rows are different hash functions. •What is a deterministic alg with a •Cols are different sets of n items to hash. good worst-case guarantee? •M(i,j) = #collisions incurred by alg i on set j. • A row that does well against all columns. We saw: •What is a lower bound for deterministic •For any row, can reverse-engineer a bad column algorithms? (if universe of keys is large enough). • Showing that for each row i there exists a column j •Universal hashing: a randomized strategy for row such that the cost M(i,j) is high. player that has good behavior for every column. •How to give lower bound for randomized – For any sequence of operations, if you randomly construct hash function in this way, you won’t get many algs? collisions in expectation. • Give randomized strategy (ideally minimax optimal) for adversary that is bad for all i. Must also be bad for all distributions over i. Sometimes called Yao’s principle. 3 Lower bounds for Lower bounds for Adversary Adversary randomized sorting randomized sorting Alg player Alg player •Adversary strategy: uniform random •Q: How many leaves at depth ≤ lg(n!)-10? permutation of {1,2,…,n} •A: At most 1+2+4+…+n!/1024 < n!/512. •Any deterministic algorithm can be viewed •So, over 99% of leaves at depth ¸ lg(n!)-10, so as a decision tree with n! leaves. No two average depth is (lg(n!)) = (n log n). input orderings can go to same leaf. x2 > x4? x2 > x4? x2 > x5? x4 > x5? x2 > x5? x4 > x5? x3 > x6? x2 > x6? x3 > x6? x2 > x5? x3 > x6? x2 > x6? x3 > x6? x2 > x5? How deep is random leaf in tree with n! leaves? How deep is random leaf in tree with n! leaves? Interesting game Interesting game “Smuggler vs border guard” “Smuggler vs border guard” • Graph G, source s, sink t. Smuggler chooses path. • Border guard: find min cut, pick random edge in it. Border guard chooses edge to watch. • Smuggler: find max flow, scale to unit flow, induces • If edge is in path, guard wins, else smuggler wins. prob dist on paths. s t s t • What are the minimax optimal strategies? • Minimax theorem ⇔ Maxflow-mincut theorem General-Sum Games General-sum games • Zero-sum games are good formalism for design/analysis of algorithms. • In general-sum games, can get win-win and lose-lose situations. • General-sum games are good models for street to drive on systems with many participants whose • E.g., “what side of sidewalk to walk on?”: person behavior affects each other’s interests Left Right walking – E.g., routing on the internet towards you Left (1,1) (-1,-1) – E.g., online auctions you Right (-1,-1) (1,1) 4 General-sum games Nash Equilibrium • A Nash Equilibrium is a stable pair of • In general-sum games, can get win-win strategies (could be randomized). and lose-lose situations. • Stable means that neither player has • E.g., “which movie should we go to?”: incentive to deviate on their own. • E.g., “what side of sidewalk to walk on”: Martian Hotel Tr Left Right The Martian (8,2) (0,0) Left (1,1) (-1,-1) Hotel Transl (0,0) (2,8) Right (-1,-1) (1,1) No longer a unique “value” to the game. NE are: both left, both right, or both 50/50. Uses • Economists use games and equilibria as Existence of NE models of interaction. • Nash (1950) proved: any general-sum game • E.g., pollution / prisoner’s dilemma: must have at least one such equilibrium. – (imagine pollution controls cost $4 but improve – Might require using randomization as in minmax. everyone’s environment by $3) • This also yields minimax thm as a corollary. don’t pollute pollute – Pick some NE and let V = value to row player in that equilibrium. don’t pollute (2,2) (-1,3) – Since it’s a NE, neither player can do better even knowing the (randomized) strategy their pollute (3,-1) (0,0) opponent is playing. – So, they’re each playing minimax optimal! Need to add extra incentives to get good overall behavior. Existence of NE Proof • Proof will be non-constructive. • We’ll start with Brouwer’s fixed point theorem.
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