Towards an evolutionary cooperative game theory
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1 Towards an evolutionary cooperative game theory Andre Casajus and Harald Wiese March 2010 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
2 Two game theories Noncooperative game theory = strategy-oriented game theory Strategies, payo functions Nash equilibrium Cooperative game theory = payo -oriented game theory Coalition functions Core, Shapley value Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
3 Two game theories but only one evolutionary game theory By evolutionary game theory, we normally understand evolutionary noncooperative game theory: Players are drawn at random from a large population. They are programmed to play a certain (mixed) strategy and the strategy that does better than other strategies grows faster i.e., more players use the successful strategies. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
4 Two game theories but only one evolutionary game theory By evolutionary game theory, we normally understand evolutionary noncooperative game theory: Players are drawn at random from a large population. They are programmed to play a certain (mixed) strategy and the strategy that does better than other strategies grows faster i.e., more players use the successful strategies. How about an evolutionary cooperative game theory? Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
5 Two game theories but only one evolutionary game theory By evolutionary game theory, we normally understand evolutionary noncooperative game theory: Players are drawn at random from a large population. They are programmed to play a certain (mixed) strategy and the strategy that does better than other strategies grows faster i.e., more players use the successful strategies. How about an evolutionary cooperative game theory? John Nash received a grant from the NSF to develop a new evolutionary solution concept for cooperative games. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
6 Evolutionary cooperative game theory overview I Idea: Agents are programmed to assume a certain player role. Agents payo > tness > proliferation (s 1 (0),..., s n (0)) > payo s > (s 1 (1),..., s n (1)) > Problem: s i (t) will not be natural numbers Solution: extended coalition function de ned for coalitions (s 1 (t),..., s n (t)) we use the Lovasz extension v ` (u`t (s) = min i2t s i ) Problem: extensions cannot be an input for the (standard) Shapley value. Solution: continuous Shapley value introduced by Aumann and Shapley (1974) which uses derivatives Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
7 Evolutionary cooperative game theory overview II Problem: Lovasz extensions v ` are not di erentiable Solution: approximation with di erentiable functions leading to continuous Shapley payo s and their limits that feed into a replicator dynamic = di erential equation which is Problem: not solvable by standard means Solution: consider discrete version of replicator dynamic, increase number of steps and decrease step length and let go towards in nity and zero, respectively. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
8 Results unanimity game for three players: the two productive players 1 and 2 win 2 time periods with step length (x 1 (0), x 2 (0), x 3 (0)) = 1 6, 2 6, 1 2! 1 2, 1 2, , , 2 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
9 Results apex game: the apex player teams up with player 4 2 time periods with step length (x 1 (0), x 2 (0), x 3 (0), x 4 (0)) = 2 10, 1 10, 3 10, 4 10! 1 2, 0, 0, , 4 2 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
10 Results apex game: the three unimportant players trump the apex player 2 time periods with step length (x 1 (0), x 2 (0), x 3 (0), x 4 (0)) = 1 10, 2 10, 3 10, 4 10! 0, 1 3, 1 3, , 3 2, 3, Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
11 Results apex game with identical starting shares 2 time periods with step length (x 1 (0), x 2 (0), x 3 (0), x 4 (0)) = 1 4, 1 4, 1 4, 1 4! 1 4, 1 4, 1 4, 1 4 1, 2, 3, 4 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
12 And now the mathematical details Coalition functions as vectors Measuring agents The Lovasz extension and its approximation The continuous Shapley value The replicator dynamics Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
13 Payo vectors A payo vector x for N is an element of R N or a function N! R. We de ne R N + := x 2 R N : x i 0 for all i 2 N, R N ++ := x 2 R N : x i > 0 for all i 2 N, := (N) := x 2 R N + : x i = 1 and int ( ) := int ( (N)) = x 2 R N ++ : x i = 1. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
14 Agents and measures intervals of agents s = (s 1,..., s n ) 2 R N + In case of s 2 f0, 1g N, we identify s with the coalition K (s) := fi 2 N : s i = 1g. λ = Lebesgues-Borel measure on R Choose n non-intersecting intervals I i R with λ (I i ) = s i I := [ i2n I i = set of all agents B = set of Borel sets of I µ s i de ned by µ s i (K ) := λ (K \ I i ), K 2 B, is a measure on (I, B). µ s = µ s i : B! R N, K 7! (µ s i (K )) i2n = Cartesian product of i2n these measures µ s (K ) distributes the agents in K among the n players (player types) and attributes a measure to each player. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
15 Andre Casajus and Harald Wiese () Towards an evolutionary Figure: cooperative game theory March / 32 Agents and measures an example N = f1, 2g s = 1 2, 2 and intervals I 1 = 0, 1 2 and I2 = [2, 4]. For K := 0, 1 4 [ 5 2, 3 2 B we obtain µ s 1 (K ) = λ (K \ I 1) = λ 0, 1 = and similarly for µs 2 I1 I K = 0,, 3 4 2
16 Lovasz extension Approximation Let s := min T (s) := min i2t s i be the minimum player size of the T -players and let T := fj 2 Njs j = s g be the set of T -players with minimal size. For 6= T N and m 2 N, we de ne u`,m T = min m T : RN +! R by 8 0, s = 0 >< min m jt T (s) := j m 1 1, else. >: m i2t 1 s m i s 2 R N +, and have De ne v `,m by v `,m (s) := lim m! minm T (s) = min T (s) m v (T ) min m T (s), 6=T N m 2 N Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
17 Vector measure games In our setting, vector measures games are given by Given a coalition K 2 B, v `,s : = v ` µ s : B! R and v `,m,s : = v `,m µ s : B! R µ s (K ) speci es how to devide K among the n groups and how to measure these subgroups. v ` or v `,m then yield the worth in accordance with the underlying TU game v. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
18 Shapley value for vector measure games Aumann/Shapley 1974 (Theorem B) = diagonal formula: Let K 2 B be any continuous coalition of agents. They receive Sh v `,m,s (K ) = n Z 1 µ s j (K ) j=1 0 v `,m s j dτ τs Aanalogue of player j s marginal contribution in the discrete Shapley formula = derivative of the coalition s worth with respect to the measure of agents of player j. Evaluation at τs = (τs 1,..., τs n ) (diagonal formula): Draw a subset of agents by chance. More likely than not, the composition in this subset (how many agents of player 1, player 2 etc.) will not deviate much from the composition in the overall population. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
19 Shapley value for the agents of player i Lemma We have 8 >< Sh u`,m,s T (I i ) = >: and Sh u`,s T 0, i /2 T 0, s = 0 jt j m 1 m+1 si m j2t sj m m, i 2 T and s 6= 0 (I i ) := lim Sh u`,m,s m! T (I i ) = ( s jt j, i 2 T, s 6= 0 0, otherwise De nition The averge payo accruing to agents from I i is also called agent i s payo and is given by Sh i v `,s := Sh v `,s (I i ) s i. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
20 Shapley value for the agents of player i example apex game Sh 1 h`,s, Sh 2 h`,s, Sh 3 h`,s, Sh 4 h`,s = 8 (0, 1, 0, 0), s 1 < s 2 < s 3 < s 4 0, 1 2, 1 2, 0 s 1 < s 2 = s 3 < s 4 0, 1 3, 1 3, 1 3 s 1 < s 2 = s 3 = s 4 1 ><, 1 2, 0, 0 s 1 = s 2 < s 3 < s 4 2 3, 1 6, 1 6, 0 s 1 = s 2 = s 3 < s 4... (0, 0, 0, 1) s 2 < s 3 < s 4 < s 1 0, 0, 1 2, 1 2 s 2 < s 3 = s 4 < s 1 >: 0, 1 3, 1 3, 1 3 s 2 = s 3 = s 4 < s 1 (0, 0, 0, 1) s 2 = s 3 < s 4 < s 1 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
21 Replicator dynamics the agents Shapley payo s = tness a constant birthrate β a constant death rate δ > evolution of s i is de ned by ṡ i = hβ + Sh i v `,s i δ s i. In terms of population shares we obtain the replicator dynamics s i x i := n j=1 s j ẋ i = Sh i v `,s n! Sh j v `,s x j x i j=1 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
22 Replicator dynamics from discrete to continuous I Existence not guaranteed by standard methods. Therefore: x i (t) = x i (t 1) + x i (t 1) " Sh i v `,x (t 1) n Sh j v `,x (t # 1) x j j=1 In order to smooth out the solution orbit, we introduce a (very small) step length σ > 0 and work with the replicator dynamics " # x i (t) = x i (t 1) + x i (t 1) σ Sh i v `,x (t 1) n Sh j v `,x (t 1) x j, t j=1 In a continuous case, σ would a ect the velocity of change but not the solution orbit. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
23 Replicator dynamics from discrete to continuous I We use the formula number of time periods = number of steps times step length De nition The Euler replicator dynamic for T time periods is de ned by the discrete replicator dynamics obeying 0 t S, σ = T S and S!. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
24 Steady states, asymptotic stability De nition A vector of population shares ˆx = (ˆx 1,..., ˆx n ) 2 is a steady state if there exists a population share vector x (0) = (x 1 (0),..., x n (0)) 2 such that the Euler replicator dynamics yields for all i = 1,..., n. De nition lim T! x i (t) = ˆx i A steady state ˆx = (ˆx 1,..., ˆx n ) is called asymptotically stable if there exists some ε > 0 such that for all population vectors x (0) obeying kx (0) ˆxk 2 < ε we have lim x (t) = ˆx. T! Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
25 Negative results Dominant players need not vanish De nition Player i 2 N strictly dominates player j 2 N if v (K [ fig) > v (K [ fjg) holds for all K Nn fi, jg. Example: N = f1, 2g, v (1) = 1, v (2) = 0 and v (1, 2) = 3 : 1 2 1, 2 Figure: Player 2 is dominated but does not vanish. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
26 Simple games De nition A game v 2 V (N) is called simple if it is monotonic, obeys v (K ) = 0 or v (K ) = 1 for every coalition K N and v (N) = 1. Set of minimal winning coalitions M Examples: Unanimity games, apex games, contradictory games (where there is a coalition K such that both K and NnK are winning coalitions) Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
27 Simple games Lovasz extension and derived simple game For s 2 R N +, we have where the arguments are M max min M and N max min N. v ` (s) = max K 2M min i2k s i. Given M and s, we de ne the derived simple game v (M, s) on the player set Nmax min by specifying: W N max min is a winning coalition if there exists a coalition K 2 M max min s.t. W = K \ Nmax min Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
28 Simple games An agent s Shapley payo = his player s Shapley payo in derived game v 2 V (N) > v (M, s) derived simple game The agents Shapley values for players i 2 N are given by Sh i v `,s Sh = i (v (M, s)), i 2 Nmax min 0, otherwise, Thus, in a simple game, a player obtains a non-zero payo zero if and only if he belongs to minimal winning coalition, his size is minimal within at least one minimal winning coalition, and this minimal size is at least as large as the minimal sizes found in any other winning coalition. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
29 Simple games Characterization of asymptotically stable states v 2 V (N) simple game with M. Asymptotically stable states ˆx = (ˆx 1,..., ˆx n ) are characterized by minimal wining coalitions W 2 M and Example: apex game ˆx i = ( 1 jw j, i 2 W 0, otherwise Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
30 Conclusions Interpretation and application ENGT s basic model: pairwise contests monomorphic population playing a symmetric game selection of equilibrium strategies ECGT basic model (as presented here): playing the eld and polymorphic selection of players and coalitions Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
31 Conclusions Filar and Petrosjan s dynamic cooperative games In our model, the agents shares change. Alternatively, players themselves could grow: Depending on their pro ts, rms grow in an organic fashion (rather than grow by mergers and acquisitions). Filar and Petrosjan (2000) present dynamic cooperative games where they de ne a sequence of games (in discrete or in continuous time) so that one TU game is determined by the previous one and by the payo s achieved under some solution concept. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
32 Conclusions Future work Selection = evolution for a given set of parameters Mutation = change of parameters We may consider small changes of the coalition function v. Other players could be added with very small sizes such that the worths for the other players stays the same for a zero size of the new arrival. Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
33 Results apex game with nearly identical starting shares 2 time periods with step length (x 1 (0), x 2 (0), x 3 (0), x 4 (0)) = ε, 1 4! 0, 1 3, 1 3, 1 3 discreteness problem! ε 3, 1 4 ε 3, 1 4 ε 3 2, 3, 4 1 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
34 Two di erent non-zero shares Consider N = f1, 2, 3g, v 2 V (N) given by v (1) = v (2) = v (3) = 0, v (1, 3) = 2, v (1, 2) = v (2, 3) = 1 and v (1, 2, 3) = , 3 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March / 32
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