Potential Game and Its Application to Control Daizhan Cheng - PowerPoint PPT Presentation

Potential Game and Its Application to Control Daizhan Cheng Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences Seminar for SJTU Combinatorics Week Shanghai Jiao Tong University Shanghai, April 27, 2015

Outline of Presentation An Introduction to Game Theory 1 Semi-tensor Product of Matrices 2 Potential Games 3 Decomposition of Finite Games 4 Networked Evolutionary Games 5 Applications 6 Conclusion 7 2 / 76

I. An Introduction to Game Theory ☞ Game Theory Figure 1: John von Neumann J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior , Princeton University Press, Princeton, New Jersey, 1944. 3 / 76

☞ Non-Cooperative Game (Winner of Nobel Prize in Economics 1994) Figure 2: John Forbes Nash Jr. J. Nash, Non-cooperative game, The Annals of Math- ematics , Vol. 54, No. 2, 286-295, 1951. 4 / 76

☞ Cooperative Game (Winner of Nobel Prize in Economics 2012 with Roth) Figure 3: Lloyd S. Shapley D. Gale, L.S. Shapley, Colle admissions and the stabil- ity of marriage, Vol. 69, American Math. Monthly, 9-15, 1962. 5 / 76

☞ Market Power and Regulation (Winner of Nobel Prize in Economics 2014) Figure 4: Jean Tirole D. Fudenberg and J. Tirole, Game Theory, MIT Press, Cam- bridge, MA, 1991. J. Tirole, The Theory of Industrial Organization, MIT Press, Cambridge, MA, 1988. 6 / 76

☞ Normal Non-cooperative Game Definition 1.1 A normal game G = ( N , S , c ) : (i) Player : N = { 1 , 2 , · · · , n } . (ii) Strategy : S i = D k i , i = 1 , · · · , n , where D k := { 1 , 2 , · · · , k } . n � (iii) Profile : S = S i . i = 1 (iv) Payoff function : c j : S → R , j = 1 , · · · , n . (1) c := { c 1 , · · · , c n } . 7 / 76

☞ Nash Equilibrium Definition 1.2 In a normal game G , a profile s = ( x ∗ 1 , · · · , x ∗ n ) ∈ S is a Nash equilibrium if c j ( x ∗ 1 , · · · , , x ∗ j , · · · , x ∗ n ) ≥ c j ( x ∗ 1 , · · · , x j , · · · , x ∗ n ) (2) j = 1 , · · · , n . 8 / 76

☞ Nash Equilibrium Example 1.3 Consider a game G with two players: P 1 and P 2 : Strategies of P 1 : D 2 = { 1 , 2 } ; Strategies of P 2 : D 3 = { 1 , 2 , 3 } . Table 1: Payoff bi-matrix P 1 \ P 2 1 2 3 6, 1 1 2 , 1 3 , 2 2 1 , 6 2 , 3 5 , 5 ( 1 , 2 ) is a Nash equilibrium. 9 / 76

☞ Mixed Strategies Definition 1.4 Assume the set of strategies for Player i is S i = { 1 , · · · , k i } . Then Player i may take j ∈ S i with probability r j ≥ 0 , j = 1 , · · · , k i , where k i � r j = 1 . j = 1 Such a strategy is called a mixed strategy . Denote by x i = ( r 1 , r 2 , · · · , r k i ) T ∈ ∆( S i ) . 10 / 76

Notations Mixed Strategy: � � k � ( r 1 , r 2 , · · · , r k ) T � � r i ≥ 0 , Υ k := r i = 1 . i = 1 Probabilistic Matrix: � � � � Col ( M ) ⊂ Υ m Υ m × n := M ∈ M m × n . ) T . 1 m := ( 1 , · · · , 1 � �� � m 11 / 76

☞ Existence of Nash Equilibrium Definition 1.5 (Nash 1950) In the n -player normal game, G = ( N , S , c ) , if | N | and | S i | , i = 1 , · · · , n are finite, then there exists at least one Nash equilibrium, possibly involving mixed strategies. 12 / 76

II. Semi-tensor Product of Matrices A m × n × B p × q =? Definition 2.1 Let A ∈ M m × n and B ∈ M p × q . Denote t := lcm ( n , p ) . Then we define the semi-tensor product (STP) of A and B as � � � � A ⋉ B := A ⊗ I t / n B ⊗ I t / p ∈ M ( mt / n ) × ( qt / p ) . (3) 13 / 76

☞ Important Comments When n = p , A ⋉ B = AB . So the STP is a generaliza- 1 tion of conventional matrix product. STP keeps almost all the major properties of the con- 2 ventional matrix product available. Associativity, Distributivity; ( A ⋉ B ) T = B T ⋉ A T ; ( A ⋉ B ) − 1 = B − 1 ⋉ A − 1 ; · · · . 14 / 76

☞ Logical Variable and Logical Matrix Vector Form of Logical Variables : x ∈ D k = { 1 , 2 , · · · , k } , we identify i ∼ δ i i = 1 , · · · , k , k , where δ i k is the i th column of I k . Then x ∈ ∆ k , where ∆ k = { δ 1 k , · · · , δ k k } . Logical Matrix : L = [ δ k 1 m , δ k 2 m , · · · , δ k n m ] , shorthand form: L = δ m [ k 1 , k 2 , · · · , k n ] . 15 / 76

☞ Matrix Expression of Logical Functions Theorem 2.1 Let x i ∈ D k i , i = 1 , · · · , n be a set of logical variables. Let f : � n i = 1 D k i → D k 0 and y = f ( x 1 , · · · , x n ) . (4) Then there exists a unique matrix M f ∈ L k 0 × k ( k = � n i = 1 k i ) such that in vector form y = M f ⋉ n i = 1 x i := M f x , (5) where x = ⋉ n i = 1 x i . M f is called the structure matrix of f , and (5) is the algebraic form of (4). 16 / 76

☞ Matrix Expression of Pseudo-logical Functions Theorem 2.1(cont’d) Let c : � n i = 1 D k i → R and h = c ( x 1 , · · · , x n ) . (6) Then there exists a unique (row) vector V c ∈ R k , such that in vector form h = V c x , (7) V c is called the structure vector of c , and (7) is the algebraic form of (6) 17 / 76

☞ Khatri-Rao Product Definition 2.2 Let A ∈ M p × m , B ∈ M q × m . Then the Khatri-Rao product of A and B is defined as M ∗ N := [ Col 1 ( M ) ⋉ Col 1 ( N ) · · · Col m ( M ) ⋉ Col m ( N )] . (8) 18 / 76

☞ Matrix Expression of Logical Mapping Let x i , y j ∈ D k , i = 1 , · · · , n , j = 1 , · · · , m , and F : D n k → D m k be y j = f j ( x 1 , · · · , x n ) , j = 1 , · · · , m . (9) Then in vector form we have j = 1 , · · · , m . y j = M j x , (10) Theorem 2.3 F can be expressed as y = M F x . (11) where y = ⋉ m j = 1 y j , and M F = M 1 ∗ M 2 ∗ · · · ∗ M m ∈ L 2 m × 2 n . (12) 19 / 76

III. Potential Games ☞ Vector Space Structure of Finite Games G [ n ; k 1 , ··· , k n ] : the set of finite games with | N | = n , | S i | = k i , i = 1 , · · · , n ; In vector form: x i ∈ S i = ∆ k i , i = 1 , · · · , n ; c i : � n i = 1 D k i → R can be expressed (in vector form) as c i ( x 1 , · · · , x n ) = V c i ⋉ n i = 1 , · · · , n , j = 1 x j , where V c i is the structure vector of c i . Set V G := [ V c 1 , V c 2 , · · · , V c n ] ∈ R nk . Then each G ∈ G [ n ; k 1 , ··· , k n ] is uniquely determined by V G . Hence, G [ n ; k 1 , ··· , k n ] has a natural vector structure as G [ n ; k 1 , ··· , k n ] ∼ R nk . 20 / 76

☞ Potential Games Definition 3.1 Consider a finite game G = ( N , S , C ) . G is a potential game if there exists a function P : S → R , called the potential function, such that for every i ∈ N and for every s − i ∈ S − i and ∀ x , y ∈ S i c i ( x , s − i ) − c i ( y , s − i ) = P ( x , s − i ) − P ( y , s − i ) , i = 1 , · · · , n . (13) D. Monderer, L.S. Shapley, Potential Games, Games and Economic Behavior , Vol. 14, 124-143, 1996. 21 / 76

☞ Fundamental Properties Theorem 3.2 If G is a potential game, then the potential function P is unique up to a constant number. Precisely if P 1 and P 2 are two potential functions, then P 1 − P 2 = c 0 ∈ R . Theorem 3.3 Every finite potential game possesses a pure Nash equilib- rium. Certain evolutions (Sequential or cascading MBRA) lead to a Nash equilibrium. D. Monderer, L.S. Shapley, Potential games, Games Econ. Theory, 97, 81-108, 1996. 22 / 76

☞ Is a Game Potential? Numerical computation ( n = 2 ): Shapley (96): O ( k 4 ) ; O ( k 3 ) ; Hofbauer (02): O ( k 2 ) ; Hilo (11): Cheng (14): Potential Equation. Hilo: “It is not easy, however, to verify whether a given game is a potential game.” D. Monderer, L.S. Shapley, Potential games, Games Econ. Theory, 97, 81-108, 1996. J. Hofbauer, G. Sorger, A differential game approach to evolutionary equilibrium selection, Int. Game Theory Rev. 4, 17-31, 2002. Y. Hino, An improved algorithm for detecting potential games, Int. J. Game Theory, 40, 199-205, 2011. D. Cheng, On finite potential games, Automatica , Vol. 23 / 76 50, No. 7, 1793-1801, 2014.

Lemma 3.4 G is a potential game if and only if there exist d i ( x 1 , · · · , ˆ x i , · · · , x n ) , which is independent of x i , such that c i ( x 1 , · · · , x n ) = P ( x 1 , · · · , x n ) (14) + d i ( x 1 , · · · , ˆ x i , · · · , x n ) , i = 1 , · · · , n , where P is the potential function. Structure Vector Express : c i ( x 1 , · · · , x n ) := V c i ⋉ n j = 1 x j V d d i ( x 1 , · · · , ˆ x i , · · · , x n ) := i = 1 , · · · , n , i ⋉ j � = i x j , P ( x 1 , · · · , x n ) V P ⋉ n := j = 1 x j . 24 / 76

Define: �� q q ≥ p j = p k j , k [ p , q ] := 1 , q < p . Construct: := I k [ 1 , i − 1 ] ⊗ 1 k i ⊗ I k [ i + 1 , n ] E i (15) ∈ M k × k / k i , i = 1 , · · · , n . Note that 1 k ∈ R k is a column vector with all entries equal 1 ; I s ∈ M s × s is the identity matrix and I 1 := 1 . � T ∈ R k n − 1 , � V d i = 1 , · · · , n . ξ i := (16) i 25 / 76

☞ Potential Equation Then (14) can be expressed as a linear system: E ξ = b , (17) where       ( V c 2 − V c 1 ) T − E 1 · · · ξ 1 E 2 0 0 − E 1 · · · ( V c 3 − V c 1 ) T  0 E 3 0   ξ 2          E =  ; ξ =  ; b =  . . . . ...  .   .   .  . . .    − E 1 · · · ( V c n − V c 1 ) T 0 0 E n ξ n (18) (17) is called the potential equation and Ψ is called the potential matrix. 26 / 76