Game Theory and its Applications to Networks - Part I: Strict Competition

Transcription

1 Game Theory and its Applications to Networks - Part I: Strict Competition Corinne Touati Master ENS Lyon, Fall 200

2 What is Game Theory and what is it for? Definition (Roger Myerson, Game Theory, Analysis of Conflicts ) Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory provides general mathematical techniques for analyzing situations in which two or more individuals make decisions that will influence one another s welfare Branch of optimization Multiple actors with different objectives Actors interact with each others Corinne Touati (INRIA) Strict Competition 2 / 44

3 Strict Competition : An Example Example 2 boxers fighting. Each of them bet $ million. Whoever wins the game gets all the money... Questions: What are the players payoff? What are the possible outputs of the game? Corinne Touati (INRIA) Strict Competition 3 / 44

4 Outline Solution Concepts 2 Solving a Game 3 Infinite Games 4 Extensions Multistage (Dynamic) Games Games with Incomplete Information 5 Conclusion Corinne Touati (INRIA) Strict Competition 4 / 44

5 Outline Solution Concepts 2 Solving a Game 3 Infinite Games 4 Extensions Multistage (Dynamic) Games Games with Incomplete Information 5 Conclusion Corinne Touati (INRIA) Strict Competition Solution Concepts 5 / 44

6 Pure Competition: Modeling Definition: Two Players, Zero-Sum Game. 2 players, finite number of actions Payoffs of players are opposite Modelization We call strategy a decision rule on the set of actions Payoffs can be represented } { by a matrix A where Player chooses i, player gets aij Player 2 chooses j player 2 gets a ij A solution point is such that no player has incentives to deviate Corinne Touati (INRIA) Strict Competition Solution Concepts 6 / 44

7 Solution of a Game What is the solution of the game Player Player ? Corinne Touati (INRIA) Strict Competition Solution Concepts 7 / 44

8 Solution of a Game What is the solution of the game Player Player ? Corinne Touati (INRIA) Strict Competition Solution Concepts 7 / 44

9 Solution of a Game Spatial Representation What is the solution of the game Player Player ? Player Player 2 Corinne Touati (INRIA) Strict Competition Solution Concepts 7 / 44

10 Solution of a Game What is the solution of the game Player Player ? Interpretation: Solution point is a saddle point Value of a game: V = min max a ij = max min a ij j i i j }{{}}{{} V + V Corinne Touati (INRIA) Strict Competition Solution Concepts 7 / 44

11 Games with no solution? Proposition: For any game, we can define: V = max min i j Proof. i, min j Example: max i a ij and V + = min j a ij min a ij j max a ij. i In general V V + V V + Corinne Touati (INRIA) Strict Competition Solution Concepts 8 / 44

12 Interpretation of V and V Interpretation : Security Strategy and Level V is the utility that Player can secure ( gain-floor ). V + is the loss-ceiling for Player 2. Corinne Touati (INRIA) Strict Competition Solution Concepts 9 / 44

13 Interpretation of V and V Player Interpretation : Security Strategy and Level V is the utility that Player can secure ( gain-floor ). V + is the loss-ceiling for Player 2. Player Player 2 Player Interpretation 2: Ordered Decision Making Suppose that there is a predefined order in which players take decisions. (Then, whoever plays second has an advantage.) V is the solution value when Player plays first. V + is the solution value when Player 2 plays first. Corinne Touati (INRIA) Strict Competition Solution Concepts 9 / 44

14 Games with more than one solution? Proposition: Uniqueness of Solution A zero-sum game admits a unique V and V +. If it exists V is unique. A zero-sum game admits at most one (strict) saddle point Proof. Let (i, j) and (k, l) be two saddle points. a ij a il a kj a kl By definition of a ij : a ij a il and a ij a kj. Similarly, by definition of a kl : a kl a kj and a kl a il Then, a ij a il a kl a kj a ij. Corinne Touati (INRIA) Strict Competition Solution Concepts 0 / 44

15 Extension to Mixed Strategies Definition: Mixed Strategy. A mixed strategy x is a probability distribution on the set of pure strategies: i, x i 0, x i = i Optimal Strategies: Player maximize its expected gain-floor with x = argmax min y xay t. Player 2 minimizes its expected loss-ceiling with y = argmin max xay t. x Values of the game: V m V m + = max x = min y min y max x xay t = max x xay t = min y min xa.j and j max A i. y t. i Corinne Touati (INRIA) Strict Competition Solution Concepts / 44

16 The Minimax Theorem Theorem : The Minimax Theorem. In mixed strategies: V m = V m + Proof. def = V m Lemma : Theorem of the Supporting Hyperplane. Let B a closed and convex set of points in R n and x / B Then, n n p,...p n, p n+ : x i p i = p n+ and y B, p n+ < p i y i. i= Proof. Consider z the point in B of minimum distance to x and consider n, i n, p i = z i x i, p n+ = z i x i x i i i i= Corinne Touati (INRIA) Strict Competition Solution Concepts 2 / 44

17 The Minimax Theorem Theorem : The Minimax Theorem. In mixed strategies: V m = V m + Proof. def = V m Lemma : Theorem of the Alternative for Matrices. Let A = (a ij ) m n Either (i) (0,..., 0) is contained in the convex hull of A.,..., A.n, e,...e m. Or (ii) There exists x,..., x m s.t. m m i, x i > 0, x i =, j,..., n, a ij x i. i= Lemma 2. Lemma 3: Let A be a game and k R. Let B the game such that i, j, b ij = a ij + k. Then V m (A) = V m (B) + k and V m + (A) = V m + (B) + k. i= Corinne Touati (INRIA) Strict Competition Solution Concepts 2 / 44

18 The Minimax Theorem Theorem : The Minimax Theorem. In mixed strategies: V m = V m + def = V m Proof. From Lemma 2, we get that for any game, either (i) from lemma 2 and V m + 0 or (ii) and V m > 0. Hence, we cannot have V m 0 < V m +. With Lemma 3 this implies that V m = V m +. Corinne Touati (INRIA) Strict Competition Solution Concepts 2 / 44

19 The Minimax Theorem - Illustration Example: ( ) Corinne Touati (INRIA) Strict Competition Solution Concepts 3 / 44

20 A Note on Symmetric Games Definition: Symmetric Game. A game is symmetric if its matrix is skew-symmetric Proposition: The value of a symmetric game is 0 and any strategy optimal for player is also optimal for player 2. Proof. Note that xax t = xa t x t = (xax t ) t = xax t = 0. Hence x, min xay t 0 and max yax t 0 so V = 0. y y If x is an optimal strategy for then 0 xa = x( A t ) = xa t and Ax t 0. Corinne Touati (INRIA) Strict Competition Solution Concepts 4 / 44

21 Outline Solution Concepts 2 Solving a Game 3 Infinite Games 4 Extensions Multistage (Dynamic) Games Games with Incomplete Information 5 Conclusion Corinne Touati (INRIA) Strict Competition Solving a Game 5 / 44

22 Dominating Strategies Definition: Dominating (pure) Strategies. We say that pure strategy i dominates strategy j for player if k, a ik a jk. A similar definition holds for dominating strategies for player 2. Example: Corinne Touati (INRIA) Strict Competition Solving a Game 6 / 44

23 Dominating Strategies Definition: Dominating (pure) Strategies. We say that pure strategy i dominates strategy j for player if k, a ik a jk. A similar definition holds for dominating strategies for player 2. Example: Corinne Touati (INRIA) Strict Competition Solving a Game 6 / 44

24 Dominating Strategies Definition: Dominating (pure) Strategies. We say that pure strategy i dominates strategy j for player if k, a ik a jk. A similar definition holds for dominating strategies for player 2. Example: Corinne Touati (INRIA) Strict Competition Solving a Game 6 / 44

25 Dominating Strategies Definition: Dominating (pure) Strategies. We say that pure strategy i dominates strategy j for player if k, a ik a jk. A similar definition holds for dominating strategies for player 2. Example: Corinne Touati (INRIA) Strict Competition Solving a Game 6 / 44

26 Dominating Strategies Definition: Dominating (pure) Strategies. We say that pure strategy i dominates strategy j for player if k, a ik a jk. A similar definition holds for dominating strategies for player 2. Example: Corinne Touati (INRIA) Strict Competition Solving a Game 6 / 44

27 Dominating Strategies Definition: Dominating (pure) Strategies. We say that pure strategy i dominates strategy j for player if k, a ik a jk. A similar definition holds for dominating strategies for player 2. Example: Corinne Touati (INRIA) Strict Competition Solving a Game 6 / 44

28 Dominating Strategies Definition: Dominating (pure) Strategies. We say that pure strategy i dominates strategy j for player if k, a ik a jk. A similar definition holds for dominating strategies for player 2. Example: Corinne Touati (INRIA) Strict Competition Solving a Game 6 / 44

29 Fictitious Play Learning each other s behavior The players play a number of times Each time, each player plays so as to maximize its expected return against its opponent s observed empirical probability distribution The empirical distribution converge to optimal strategies. Example (Rock-Paper-Scissor) Paper Rock Scissor Corinne Touati (INRIA) Strict Competition Solving a Game 7 / 44

30 Linear Programming Problem for player : Maximize its gain-floor, i.e. max min a pi x i We re-write this as a linear program: x p i x i 0, i n max v s.t. x i = v x,v i= n v a ij x i, j y i= The dual problem is: y j 0, j m y j = v min v s.t. j= y,v m v a ij y j, i x j= which is the optimization problem of the second player! Corinne Touati (INRIA) Strict Competition Solving a Game 8 / 44

31 Linear Programming Example and Geometrical Interpretation: In pure strategies: V V + In mixed strategies: V = V + = 2.5, x opt = (0.5, 0.5), y opt = (0.25, 0.75) y + 3y 2 = 2.5 V y y 4y + 2y 2 = 2.5 y + y 2 = Corinne Touati (INRIA) Strict Competition Solving a Game 8 / 44

32 Outline Solution Concepts 2 Solving a Game 3 Infinite Games 4 Extensions Multistage (Dynamic) Games Games with Incomplete Information 5 Conclusion Corinne Touati (INRIA) Strict Competition Infinite Games 9 / 44

33 Infinite Games If the set of actions is infinite, the value V m may not exists, and even if it does, there may not be a solution strategy. Example 2 0 /2 /2 /3 2/3 /4 3/4 /5 4/5 For the truncated game, the solution is: (x = k 2 3k 2, x k = 2k 3k 2 ), (y = k 3k 2, y 2 = 2k 3k 2 ) with the game value V m = 2 k 3k 2. The value of the infinite game is V = 2/3 which cannot be attained. Yet, the players can secure a value arbitrarily close to. Definition: ε-saddle point. For a given ε 0, the pair (x ε, y ε ) U U 2 is called an ε-saddle point if J(x, y ε ) ε J(x ε, y ε ) J(x ε, y) + ε for all (x, y) U U 2 Corinne Touati (INRIA) Strict Competition Infinite Games 20 / 44

34 Infinite Games If the set of actions is infinite, the value V m may not exists, and even if it does, there may not be a solution strategy. Example 2 0 For the truncated game, the solution is: (x = Theorem 2: k 2Finite /2 /2 3k 2, x Value k = 2k 3k 2 ), of (y Infinite = k Game. 3k 2, y 2 = 2k 3k 2 ) with the An infinite game has a finite value /3 2/3 game value V m = 2 k 3k 2. if and only if, ε > 0, an ε-saddle point exists. /4 3/4 The value of the infinite game is V = 2/3 which /5 4/5 cannot be attained. Yet, the players can secure a value arbitrarily close to. Definition: ε-saddle point. For a given ε 0, the pair (x ε, y ε ) U U 2 is called an ε-saddle point if J(x, y ε ) ε J(x ε, y ε ) J(x ε, y) + ε for all (x, y) U U 2 Corinne Touati (INRIA) Strict Competition Infinite Games 20 / 44

35 Continuous Games Definition: Continuous Game. The strategy set is [0, ] [0, ]. The payoff function is A : [0, ] [0, ] R. A is sometimes called the kernel. Definition: Mixed Strategy. A probability distribution over the set of pure strategies. Can be represented by the cumulative distribution function F, continuous, non-decreasing, with F : [0, ] [0, ], F (0) = 0, F () =. Expected payoff for pure strategy x for player : E(x, G) = 0 A(x, y)dg(y) Expected payoff for pure strategy y for player 2: E(F, y) = 0 0 Corinne Touati (INRIA) Strict Competition Infinite Games 2 / 44 0 A(x, y)df (x) Value of the game: V m = sup inf E(F, y) and V m F y + = inf sup E(x, G) G x Expected reward E(F, G) = A(x, y)df (x)dg(y)

36 Properties Theorem 3. If A is continuous, then the forms sup inf and inf sup may be replaced by max min and min max. Proof. y E(F, y) is continuous over a compact (the interval [0, ]). By definition of V, there exists F n s.t. min E(F y n, y) > V /n. As the set of functions from [0 : ] to itself is compact, there exists a convergent subsequence of F. The limit F 0 can be extended to a continuous function attaining maximum V. Theorem 4. If A is continuous, then V = V +. Proof. Consider the sequence of matrices A n, with i, j, a n ij = A(i/n, j/n). It has a value and optimal strategies. From the uniform continuity of function A over [0, ] [0, ], the value of the continuous game is the limit of the value of the sequence of finite games. Corinne Touati (INRIA) Strict Competition Infinite Games 22 / 44

37 Concave-Convex Games Definition: Concave-Convex Games. A game is said concave-convex it y, x A(x, y) is concave and x, y A(x, y) is convex. Proposition: A continuous concave-convex game always have pure strategy solutions. Proof.... Corinne Touati (INRIA) Strict Competition Infinite Games 23 / 44

38 Outline Solution Concepts 2 Solving a Game 3 Infinite Games 4 Extensions Multistage (Dynamic) Games Games with Incomplete Information 5 Conclusion Corinne Touati (INRIA) Strict Competition Extensions 24 / 44

39 Outline Solution Concepts 2 Solving a Game 3 Infinite Games 4 Extensions Multistage (Dynamic) Games Games with Incomplete Information 5 Conclusion Corinne Touati (INRIA) Strict Competition Extensions 25 / 44

40 Multiple Round Games Simplest case of dynamic games Given number of rounds Ex: TicTacTo Resolution techniques: Backward Induction (exact solution) Behavioral strategy: collection of probability distributions for each possible information set (approximate solution) Corinne Touati (INRIA) Strict Competition Extensions 26 / 44

41 Multiple Round Games Definition: Strategies. In a game in extensive form, a strategy for a player is a sequence of actions. Actions are different from strategies. Definition: Behavorial Strategies. N i is the set of decision nodes for player i. A behavioral strategy for player i is a mapping from each node in N i to the set of (probability distributions on the) possible actions. Proposition: Existence of Solution Any zero-sum game in extensive form (finite, with full information, and without chance move) admits a saddle point in behavioral strategies. It is also a saddle point in mixed strategies. The behavioral strategy saddle point can be found recursively. Corinne Touati (INRIA) Strict Competition Extensions 27 / 44

42 Multiple Round Games Example: Player Player Strategies for player are {{}, {2}, {3}} Strategies for player 2: {,, }, {,, 2}, {,, 3}, {2,, }... (overall: 27 pure strategies). Corresponding normal form game (partial -without action 3 for player -, for display reasons ): Behavioral Strategies: For player one: a choice of p, p 2, p 3 (with p + p 2 + p 3 = ) For player two: a choice of q, q 2, q 3, q 2, q 2 2, q 2 3, q 3, q 3 2, q 3 3, with q + q 2 + q 3 =, q 2 + q q 2 3 =, q 3 + q q 3 3 = Corinne Touati (INRIA) Strict Competition Extensions 28 / 44

43 Multiple Round Games Extensions Games with Repeated Decisions There exists two (distinct) states of the game with identical (chosen) action. Extensive Forms with Cycles There exist some cycles in the state graph. Games with Partial Information The players do not have perfect information about each other s actions. Information Set depending on the Actions The knowledge of the system for a player depends on its actions. Corinne Touati (INRIA) Strict Competition Extensions 29 / 44

44 Stochastic Games There exists p states plus a state 0 representing the end of the game. In each state k, a game is played, characterized by matrix A k in R m k,n k and a matrix of probability vectors (q k ) i mk, j n k over the set of states. The matrix game is (by a great abuse of notations) α k ij = a k ij + p 0 q kl ij S l with p 0 q kl ij =, q l ij 0, q k0 ij > 0 m k A strategy vector is i= x kl i =, x kl i 0 Corinne Touati (INRIA) Strict Competition Extensions 30 / 44

45 Stochastic Games Example ( 3 + S4/2 ) ( 3 2 ) A = + A 2 = 2 S S A 3 = 2 S S + A 4 = 2 S2 2 S S2 2 x y ( ) X 3 Y 2 x 2y 2 2 x2 2y x3 2y X 2 ( ) 2 (x3 2y 3 + x 3 y 3 2) Y 2 0 X 3 ( (x4 2y 4 + x 4 y2) 4 ( ) X By considering only stationary strategies, we get that (v, B) is the fixed point solution of: p v k = val(b k ) and b k ij = a k ij + qij kl v l. Corinne Touati (INRIA) Strict Competition Extensions 3 / 44 ) Y 3 Y 4 0

46 Stochastic Games Example ( 3 + S4/2 ) ( 3 2 ) A = + A 2 = 2 S S A 3 = 2 S S + A 4 = 2 S2 2 S S2 Solving: v 0 = (0, (( 0, 0, 0) 3 B 0 = ) ( 3 2, 2 ) ( 2 2, 2 ) ( 2, 2 )) v = ( 3, 8, 2 7, 2 ) B = ( ) , 6, , 4 Corinne Touati (INRIA) Strict Competition Extensions 3 /

47 Recursive Games Extension of stochastic games where the probability of infinite play is positive. The payoff is obtained only when the game terminates. There exists also a payoff for infinite game. The matrix game is p αij k = qij k0 a k ij + qij kl S l with A strategy vector is m k i= x kt i p 0 q kl ij =, q kl ij 0 =, x kt i 0 The value iteration method can be used: the error does not vanish to 0. There may not be an optimal strategy, but only ε-optimal strategy. A A A Example: A A v =, ε-optimal strategy for player : A (0, δ δ 2, δ, δ 2 ) Corinne Touati (INRIA) Strict Competition Extensions 32 / 44

48 Differential Games Limit case of a stochastic or recursive game where time interval between stages vanishes. State space x continuous in time (of dimension n) Player chooses φ, Player 2 chooses ψ The system evolves according to x = f(x, φ, ψ) (kinematic equations) The game stops either when x attains a given closed subset of R n or at given time epoch T. The payoff is either a function of the terminal state x(t ) or an integral: T 0 G(x)dt. This kind of problems have been studied widely in the domain of optimal control theory. Corinne Touati (INRIA) Strict Competition Extensions 33 / 44

49 Differential Games Example: Robust Control in D.T.N State of the system (x, y): x mobiles that have a file (y mobiles do not) the source is in contact with mobiles without a file at rate η mobiles join the system at a rate λ mobiles with the file die at rate { νx xt = η State evolves according to: t y t ν t x t y t = η t y t + λ t Player (source) chooses η, player 2 (nature) chooses ν. Corinne Touati (INRIA) Strict Competition Extensions 34 / 44

50 Outline Solution Concepts 2 Solving a Game 3 Infinite Games 4 Extensions Multistage (Dynamic) Games Games with Incomplete Information 5 Conclusion Corinne Touati (INRIA) Strict Competition Extensions 35 / 44

51 Games with Chance Move i = 2 chance moves No user knows the output of the chance move. Then V = Only first player knows, and he plays first 2 Both users know the output of the chance move: V =.5 Corinne Touati (INRIA) Strict Competition Extensions 36 / 44

52 Games with Chance Move i = 2 chance moves No user knows the output of the chance move. Then V = Only first player knows, and he plays first 2 Both users know the output of the chance move: V =.5 Equivalent game: (of size n i n m ) A A, A A A A, B A B A, A A B A, B A A, A A, B B, A B, B Then, V = 2.5. Optimal strategy: x = (0, 0, 0, ), (i.e. player does not reveal any info) and y = (ŷ, 0, ŷ, 0) with ŷ { 2, 2 3 Corinne Touati (INRIA) Strict Competition Extensions 36 / 44 }.

53 Games with Chance Move max v +v i = 2 chance moves No user knows the output of the chance move. Then V = Only first player knows, and he plays first 2 Both users know the output of the chance move: V =.5 In behavioral strategies: { p = P rob(take action A Chance move is ), Player chooses p 2 = P rob(take action A Chance move is 2).. { q = P rob(take action A Player plays A), Player 2 chooses q 2 = P rob(take action A Player plays B).. Player : v 2p + 0.5p 2 v 0.5p +.5p 2 s.t. v 2.5( p ) + ( p 2 ) v 2 2.5( p 2 ) 0 p, p 2 min v +v 2 Player 2: v 2q + 0.5( q ) v.5q 2 s.t. v 2 0.5q +.5( q ) v 2 q ( q 2 ) 0 q, q 2 Corinne Touati (INRIA) Strict Competition Extensions 36 / 44

54 Games with Chance Move i = 2 chance moves No user knows the output of the chance move. Then V = Only first player knows, and he plays first 2 Both users know the output of the chance move: V =.5 In behavioral strategies: { p = P rob(take action A Chance move is ), Player chooses p 2 = P rob(take action A Chance move is 2).. { q = P rob(take action A Player plays A), Player 2 chooses q 2 = P rob(take action A Player plays B).. Solution: V = 2.5 p = p 2 = 0 q 2 q /3,.5q 2 q Corinne Touati (INRIA) Strict Competition Extensions 36 / 44

55 Information Sets without Chance Moves Definition: Extensive Form Game. An extensive form game is a finite tree structure with: A vertex indicating the starting point of the game, A pay-off function assigning a real number to each terminal vertex of the tree, A partition of the nodes of the tree into two player sets (with N i the set of player i), A subpartition of each player set N i into information sets η i j such that all nodes of a information set has the same number of children and that no node follows another node of the same information set. Corinne Touati (INRIA) Strict Competition Extensions 37 / 44

56 Information Sets without Chance Moves Example Full Information No Information Player Player Player Player Partial Information Player Player Definition: Behavioral Strategy. A strategy for player i is a mapping that assigns an action (resp. a distribution probability over the actions) to each information set. Corinne Touati (INRIA) Strict Competition Extensions 38 / 44

57 Information Sets without Chance Moves Properties Definition: Feedback Games. A game in extensive form is a feedback game is (i) each player has perfect information of the current level of play (ii) each player knows the state of the game at every level of play. Proposition: Solution of Feedback Games Every finite feedback game admits a saddle point in behavioral strategies. Level Level The behavioral solution strategy can be obtained using simple recursive procedures (by solving a number of normal form games). Corinne Touati (INRIA) Strict Competition Extensions 39 / 44

58 Information Sets without Chance Moves General Case Proposition: Any two-person zero-sum finite game in extensive form admits a saddle-point in mixed strategies. (But not necessarily in behavioral strategies.) Example (open-loop game): Level Level The solution is γ = { } LL with proba 3/5 RR with proba 2/5 { } LL with proba 4/5 γ 2 =. RR with proba /5 Corinne Touati (INRIA) Strict Competition Extensions 40 / 44

59 Information Sets without Chance Moves Additional Results Proposition: In games where each player recall all their past actions but are ignorant of the actions of their opponent admits a solution in behavioral strategies. Proposition: Every finite game admits a saddle point in randomized strategies. (A randomized strategy is a probability distributions over the (possibly mixed strategy) behavioral strategies.) Corinne Touati (INRIA) Strict Competition Extensions 4 / 44

60 Games in Extensive form with Chance Moves Definition: Games with chance move & partial information. Can be seen as a 3 player game with the extra player ( nature ) having a fixed mixed strategy. Example Nature Such games admit a mixed strategy equilibrium. There is no systematic way to solve them. Corinne Touati (INRIA) Strict Competition Extensions 42 / 44

61 Outline Solution Concepts 2 Solving a Game 3 Infinite Games 4 Extensions Multistage (Dynamic) Games Games with Incomplete Information 5 Conclusion Corinne Touati (INRIA) Strict Competition Conclusion 43 / 44

62 Conclusion Zero-Sum Games Two-player Zero-sum games are games of pure competition A solution point (saddle point) of the game is such that no player has incentive to deviates from. The value of the game is the corresponding payoff for player. Basic Results In games with perfect information and no chance moves: Finite zero-sum games always admit a value and solution point(s) in mixed-strategies. Infinite zero-sum games with continuous payoff have a value. If the payoff function is further concave-convex, then it also have a solution point in pure strategies. The players interests can be seen as two optimization problems that are dual from each other. Corinne Touati (INRIA) Strict Competition Conclusion 44 / 44

63 Conclusion Zero-Sum Games Two-player Zero-sum games are games of pure competition A solution point (saddle point) of the game is such that no player has incentive to deviates from. The value of the game is the corresponding payoff for player. Extensions Extensions of zero-sum games include Multistage Games: the game is repeated over time Games with Chance Move: chance is modeled as an extra player with known fixed strategy Incomplete Information Games: where players have partial information about the system actual state or each other s actions. Corinne Touati (INRIA) Strict Competition Conclusion 44 / 44