Coalgebra of market games
Transcript
Coalgebra of market games
Coalgebra of market games D. Pavlovic Games Coalgebra of market games Equilibria Positions Conclusion Dusko Pavlovic Kestrel Institute and Oxford University CALCO, Udine September 2009 Outline Coalgebra of market games D. Pavlovic Games Equilibria Coalgebra of games Positions Conclusion Equilibrium programming Position analysis Conclusion Outline Coalgebra of market games D. Pavlovic Games Equilibria Coalgebra of games Positions Conclusion Equilibrium programming Position analysis Conclusion A process is a coalgebra Coalgebra of market games D. Pavlovic Games Equilibria X R / A ⇒ M(B × X ) Positions Conclusion where ◮ A — controls ◮ B — measurements ◮ X — states A process is a coalgebra Coalgebra of market games D. Pavlovic Games Equilibria A×X R / M(B × X ) Positions Conclusion where ◮ A — controls ◮ B — measurements ◮ X — states A process is a coalgebra Coalgebra of market games D. Pavlovic Games Equilibria A×X R /B×X Positions Conclusion where ◮ A — controls ◮ B — measurements ◮ X — states Coalgebra of market games Feedback and stable control D. Pavlovic Games Equilibria A×X R /B×X B×X φ /A Positions Conclusion A×X X R γ=FixA (φ◦R) hγ,idi _ A×X φ /B×X R /A /A O _φ /B×X Coalgebra of market games A game is a coalgebra D. Pavlovic Games Equilibria A×X ̺ /B×X Positions Conclusion where ◮ A= ◮ B= ◮ X= ◮ n = {0, 1, . . . , n − 1} — players Q i∈n Ai Q i∈n Bi Q i∈n Xi — moves — values — positions Coalgebra of market games A game is a coalgebra D. Pavlovic Games Equilibria A×X ̺ /B×X Positions Conclusion where ◮ A= ◮ B= ◮ X= ◮ n = {0, 1, . . . , n − 1} — players Q i∈n Ai Q i∈n Bi Q i∈n Xi — moves — values (ordered ring) — positions Response strategy and equilibrium Coalgebra of market games D. Pavlovic Games Equilibria ̺ A×X /B×X Positions Conclusion RS i A−i × Xi / Ai RS=hRS i ◦πi ii∈m A×X RS • =Fix A (RS) X PPPP PPP PP hRS• ,idi PPP' A×X / o7 A ooo / o o ooo ooo RS where A−i = /A Y k ∈n k ,i Ak Tasks for coalgebra of games Coalgebra of market games D. Pavlovic Games Equilibria ◮ analyze positions as states in X Positions Conclusion ◮ coalgebra homomorphisms between games ◮ position bisimilarity Coalgebra of market games Tasks for coalgebra of games D. Pavlovic Games Equilibria ◮ Positions analyze positions as states in X Conclusion ◮ ◮ coalgebra homomorphisms between games ◮ position bisimilarity construct equilibria as fixed points in A RS • / A or position-wise X ◮ static 1 ◮ equilibrium at a stationary position 1 RS • /A hRS • ,xi / A×X Coalgebra of market games Tasks for coalgebra of games D. Pavlovic Games Equilibria ◮ Positions analyze positions as states in X Conclusion ◮ ◮ coalgebra homomorphisms between games ◮ position bisimilarity construct equilibria as fixed points in A RS • / A or position-wise X ◮ static 1 ◮ equilibrium at a stationary position 1 RS • /A hRS • ,xi / A×X Outline Coalgebra of market games D. Pavlovic Games Equilibria Coalgebra of games Positions Conclusion Equilibrium programming Position analysis Conclusion Coalgebra of market games Standard notions (Static: X = 1) D. Pavlovic Games A1 - Best response strategy RS = BR Equilibria Positions Conclusion s−i BRi si ⇐⇒ ∀ti ∈ Ai . ̺i (ti , s−i ) ≤ ̺i (si , s−i ) Coalgebra of market games Standard notions (Static: X = 1) D. Pavlovic Games A1 - Best response relation Equilibria Positions Conclusion s BR t ⇐⇒ ∀i ∈ n. s−i BRi ti Coalgebra of market games Standard notions (Static: X = 1) D. Pavlovic Games A2 - Nash equilibrium Equilibria Positions Conclusion BR • s ⇐⇒ s BR s ⇐⇒ ∀i ∈ n. s−i BRi si Coalgebra of market games Standard notions (Static: X = 1) D. Pavlovic Games A3 - Rationalizable (undominated) profile Equilibria Positions Conclusion BR ∗s ⇐⇒ ∃t. BR ∗ t ∧ tBRs Standard notions Coalgebra of market games D. Pavlovic Upshot Games Equilibria Positions Nash equilibrium is ◮ a joint result of individual optimizations ◮ a social solution of a distributed problem ◮ noone can improve their gain on their own ◮ it leads beyond the "zero sum" view of the world Conclusion Standard notions Coalgebra of market games D. Pavlovic Games Issues ◮ existence of equilibrium ◮ ◮ Q: Is BR • empty? equilibrium selection ◮ ◮ Equilibria Q: What if s, t ∈ BR • , and i plays si and j plays tj ? social benefit of equilibrium ◮ Q: Is P i ̺iB (s) a global maximum (Pareto optimal)? Positions Conclusion Standard notions Coalgebra of market games D. Pavlovic Games Issues ◮ existence of equilibrium ◮ ◮ A: Kakutani theorem equilibrium selection ◮ ◮ Equilibria A: attractor dynamics social benefit of equilibrium ◮ A: program the notions of response Positions Conclusion Coalgebra of market games Example: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) D. Pavlovic Games For Equilibria ◮ i ∈ 2 = {0, 1} ◮ Ai = {retreat, attack} ◮ Bi = R the payoff A0 × A1 Positions ̺ Conclusion / B0 × B1 is given by retreat attack 0 w 2 retreat w 2 w w 2 w attack 0 w 2 −c −c Coalgebra of market games Example: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) D. Pavlovic Games Equilibria Positions If c < w 2, the only Nash equilibrium 1 s s = hattack, attacki with the social gain of X ̺(s) = w − 2c Conclusion / A0 × A1 is Example: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) Coalgebra of market games D. Pavlovic Games Equilibria Positions Conclusion The unstable profile r = hretreat, retreati would give the social gain of X ̺(s) = w Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games B1 - Stable strategy RS = SR Equilibria Positions Conclusion s−i SRi si ⇐⇒ ∀t ∈ A ∀ε > 0. (1 − ε)̺i (ti , s−i ) + ε̺i (ti , t−i ) ≤ (1 − ε)̺i (si , s−i ) + ε̺i (si , t−i ) Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games B1 - Stable strategy RS = SR Equilibria Positions Conclusion s−i SRi si ⇐⇒ ∀ti ∈ Ai . ̺i (ti , s−i ) ≤ ̺i (si , s−i ) ∧ (̺i (ti , s−i ) = ̺i (si , s−i ) ⇒ ∀t−i ∈ A−i . ̺i (ti , t−i ) ≤ ̺i (si , t−i )) Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games B1 - Stable response relation Equilibria Positions Conclusion s SR t ⇐⇒ ∀i ∈ n. s−i SRi ti Static strategies (X = 1) Coalgebra of market games D. Pavlovic Games B2 - Stable equilibrium Equilibria Positions Conclusion SR • s ⇐⇒ ∀i ∈ n. s−i SRi si ⇐⇒ ∀t ∈ A.∀i ∈ n. ̺i (ti , s−i ) ≤ ̺i (si , s−i ) ∧ ̺i (si , t−i ) = ̺i (si , s−i ) ⇒ ∀t−i ∈ A−i . ̺i (ti , t−i ) ≤ ̺i (si , t−i ) Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games B3 - Stable (admissible) profile Equilibria Positions Conclusion SR ∗ s ⇐⇒ ∃t.SR ∗ t ∧ tSRs Coalgebra of market games Example: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) D. Pavlovic Games Equilibria Positions If c < w 2, the only stable profile 1 s Conclusion / A0 × A1 is s = hattack, attacki with the social gain of X ̺(s) = w − 2c Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games C1 - Uniform strategy RS = UR Equilibria Positions Conclusion s−i URi si ⇐⇒ s−i BRi si ∧ ∀t−i ∈ A−i . si BR−i t−i ⇒ t−i BRi si Static strategies (X = 1) Coalgebra of market games D. Pavlovic Games C2 - Uniform equilibrium Equilibria Positions Conclusion UR • s ⇐⇒ ∀i ∈ n. s−i URi si ⇐⇒ BR • s ∧ ∀i ∈ n. ∀t−i ∈ A−i . si BR−i t−i ⇒ t−i BRi si Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games C3 - Uniform profile Equilibria Positions Conclusion UR ∗ s ⇐⇒ ∃t.UR ∗ t ∧ tURs Example: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) Coalgebra of market games D. Pavlovic Games Equilibria Positions Conclusion If c < w2 , there is no uniform equilibrium for Bird Politics (neither pure nor mixed). Example: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) Coalgebra of market games D. Pavlovic Games Equilibria Positions Conclusion If c < w2 , there is no uniform equilibrium for Bird Politics (neither pure nor mixed). The Kakutani theorem does not apply because the uniform response relation is not convex. Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games D1 - Constructive strategy RS = CR Equilibria Positions Conclusion s−i CRi si ⇐⇒ ∀ti ∈ Ai . ̺i (si , s−i ) < ̺i (ti , s−i ) ⇒ ∃t−i ∈ A−i . ̺−i (ti , t−i ) > ̺−i (ti , s−i ) ∧ ̺i (ti , t−i ) < ̺i (si , s−i ) Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games D2 - Constructive equilibrium Equilibria Positions Conclusion CR • s ⇐⇒ ∀i.s−i CRi si Coalgebra of market games Static strategies (X = 1) D. Pavlovic Games D3 - Uniform profile Equilibria Positions Conclusion CR ∗ s ⇐⇒ ∃t.CR ∗ t ∧ tCRs Example: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) Coalgebra of market games D. Pavlovic Games Equilibria Positions Conclusion There is a constructive equilibrium, consisting of the mixed strategies favoring the Pareto optimal solution hretreat, retreati. Outline Coalgebra of market games D. Pavlovic Games Coalgebra of games Equilibria Positions Coordination Competition Equilibrium programming Position analysis Problem of coordination Problem of competition Conclusion Conclusion Coalgebra of market games Tasks for coalgebra of games D. Pavlovic Games Equilibria ◮ Positions analyze positions as states in X Coordination Competition ◮ ◮ coalgebra homomorphisms between games ◮ position bisimilarity construct equilibria as fixed points in A RS • / A or position-wise X ◮ static 1 ◮ equilibrium at a stationary position 1 RS • /A hRS • ,xi / A×X Conclusion Types-as-positions: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) For Coalgebra of market games D. Pavlovic Games Equilibria ◮ i ∈ 2 = {0, 1} ◮ Ai = {retreat, attack} ◮ ◮ Positions ◮ x0 = Prob(a1 = retreat) x1 = Prob(a0 = retreat) the payoff Ai × Xi Competition Conclusion Bi = R Xi = [0, 1] ◮ Coordination ̺iB / Bi is given by w 2 w + 2c w − 2c + ̺iB (attack, xi ) = xi 2 2 ̺iB (retreat, xi ) = xi Types-as-positions: Bird Politics (Chicken, Missile Crisis, Prisoners’ Dilemma. . . ) Coalgebra of market games D. Pavlovic Games Equilibria Positions Coordination Competition Conclusion [. . . ] Example: Majority game (static: X = 1) Coalgebra of market games D. Pavlovic Games Equilibria For Positions Coordination Competition ◮ i ∈ 2m + 1 = {0, 1, . . . 2m} — players ◮ Ai = {◭, ◮} — moves ◮ Bi = {0, 1} — values the payoff A ̺ / B is 1 ̺ (s) = 0 i if #{j | sj = si } > m otherwise Conclusion Coalgebra of market games Example: Majority game (static: X = 1) D. Pavlovic Games The equilibrium 1 s Equilibria / A consists of the mixed strategies Positions Coordination Competition si = 1 1 ◭+ ◮ 2 2 Conclusion with the expected social gain of E X 2m+1 X ̺(s) = 2 k =m+1 = m+ 1 2 2m+1 k k 2m+1 2 Positions for coordination: Majority game (with 1-step memory) Coalgebra of market games D. Pavlovic Games For Equilibria ◮ i ∈ 2m + 1 = {0, 1, . . . 2m} — players ◮ Ai = {◭, ◮} — moves ◮ Bi = {0, 1} — values Xi = {◭, ◮} — positions Positions Coordination Competition ◮ ◮ ξ∈X = the game A × X ̺iB (s, x) Q Conclusion Xi — initialized randomly ̺ / B × X becomes 1 = 0 ̺iX (s, x) = ^ if #{j | sj = si } > m otherwise such that #{j | sj = ^} > m Positions for coordination: Majority game Coalgebra of market games D. Pavlovic Games Equilibria The equilibria X s / A are the coordination policies si (x) = x si′ (x) = ¬x which assure the social gain of X ̺(s) = 2m + 1 Positions Coordination Competition Conclusion Example: Minority game (static: X = 1) Coalgebra of market games D. Pavlovic Games Equilibria For Positions Coordination Competition ◮ i ∈ 2m + 1 = {0, 1, . . . 2m} — players ◮ Ai = {◭, ◮} — moves ◮ Bi = {0, 1} — values the payoff A ̺ / B is 1 ̺ (s) = 0 i if #{j | sj = si } ≤ m otherwise Conclusion Coalgebra of market games Example: Minority game (static: X = 1) D. Pavlovic Games Equilibria s The static equilibrium 1 / A is again Positions Coordination Competition si = 1 1 ◭+ ◮ 2 2 Conclusion with the expected social gain of E X m X ̺(s) = k =1 = m 2 2m+1 k 22m+1 k Coalgebra of market games Example: Minority game (static: X = 1) D. Pavlovic Games Equilibria Positions Coordination Competition ◮ Even with the maximal gain of ̺(s) = m, there are m + 1 players with ̺i (s) = 0. P Conclusion Coalgebra of market games Example: Minority game (static: X = 1) D. Pavlovic Games Equilibria Positions Coordination Competition ◮ Even with the maximal gain of ̺(s) = m, there are m + 1 players with ̺i (s) = 0. ◮ There is always a majority with an incentive to disturb the current state. P Conclusion Coalgebra of market games Example: Minority game (static: X = 1) D. Pavlovic Games Equilibria Positions Coordination Competition ◮ Even with the maximal gain of ̺(s) = m, there are m + 1 players with ̺i (s) = 0. ◮ There is always a majority with an incentive to disturb the current state. ◮ This leads from equilibrium to evolution. P Conclusion Positions for stabilization: Minority game (with ℓ-step memory and d ideas) Coalgebra of market games D. Pavlovic Games Suppose that the player i sees the positions from Equilibria Positions Xi = M ×S 1+d ×ℓ Coordination Competition Conclusion where ◮ M = S = {◭, ◮}ℓ — memory, strategies, ideas ◮ d = {0, 1, . . . , d − 1} — number of ideas ◮ ℓ = {0, 1, . . . , ℓ − 1} — length of the memory Positions for stabilization: Minority game (with ℓ-step memory and d ideas) Coalgebra of market games D. Pavlovic Games A position Equilibria Positions i0 i1 id xi = hµ, σ , σ , . . . , σ , ki ∈ M × S 1+d × ℓ = Xi Coordination Competition Conclusion records ◮ µ — the recent ℓ minority (winning) choices ◮ σi0 — i’s current strategy (ℓ-tuple of choices) ◮ σi1 , σi2 , . . . , σid — i’s bag of ideas for strategies ◮ k — the current moment in the ℓ-cycle history Positions for stabilization: Minority game Coalgebra of market games D. Pavlovic Games For Equilibria ◮ i ∈ 2m + 1 = {0, 1, . . . 2m} — players ◮ Ai = {◭, ◮} — moves ◮ Bi = {0, 1} — values Xi = M × S 1+d × ℓ — positions Positions Coordination Competition ◮ ◮ Conclusion ξi ∈ Xi — initialized randomly the payoff A × X ̺iB (s, x) ̺B / B remains 1 = 0 if #{j | sj = si } ≤ m otherwise Positions for stabilization: Minority game . . . while the position update A × Xi ̺iX Coalgebra of market games D. Pavlovic / Xi maps Games Equilibria ̺iX (s, i∗ i∗ hµ, σ , ki) = hµ̃, σ̃ , k̃i Positions Coordination Competition so that Conclusion ◮ k̃ = k + 1 mod ℓ ◮ µ̃ = h^, µ0 , µ1 , . . . , µℓ−2 i ◮ ◮ where ^ is the minority choice, i.e. #{j | sj = ^} ≤ m σ̃i∗ is obtained by reordering ◮ σ̂i∗ = hσi∗ , σi∗ , σi∗ , . . . , σi∗ i ℓ−1 0 1 ℓ−2 ◮ to maintain the invariant ∆(σ̃i0 , µ̃) ≤ ∆(σ̃i1 , µ̃) ≤ · · · ≤ ∆(σ̃id , µ̃) Positions for stabilization: Minority game . . . while the position update A × Xi ̺iX Coalgebra of market games D. Pavlovic / Xi maps Games Equilibria ̺iX (s, i∗ i∗ hµ, σ , ki) = hµ̃, σ̃ , k̃i Positions Coordination Competition so that Conclusion ◮ k̃ = k + 1 mod ℓ ◮ µ̃ = h^, µ0 , µ1 , . . . , µℓ−2 i ◮ ◮ where ^ is the minority choice, i.e. #{j | sj = ^} ≤ m σ̃i∗ is obtained by reordering ◮ σ̂i∗ = hσi∗ , σi∗ , σi∗ , . . . , σi∗ i ℓ−1 0 1 ℓ−2 ◮ to maintain the invariant ∆(σ̃i0 , µ̃) ≤ ∆(σ̃i1 , µ̃) ≤ · · · ≤ ∆(σ̃id , µ̃) — thus σ̃i0 is the best and σ̃id the worst strategy w.r.t. µ̃ Positions for stabilization: Minority game Coalgebra of market games D. Pavlovic Games Equilibria Positions Coordination Competition Let the profile X s / A be defined by si (µ, σi∗, k) = σi0 k i.e., each player plays his currently best strategy. Conclusion Positions for stabilization: Minority game Coalgebra of market games D. Pavlovic Games Equilibria Evolution: refine A × Xi ̺iX Positions / X i Coordination Competition Conclusion ◮ Each player randomly mutates her state by ◮ dropping her worst idea σi(ℓ−1) ∈ {◭, ◮}ℓ ◮ adding a random idea σ′ ∈ {◭, ◮}ℓ . at chosen intervals, or triggered by bad scores. Positions for stabilization: Minority game Coalgebra of market games D. Pavlovic Games Equilibria Evolution: refine A × Xi ̺iX Positions / X i Coordination Competition Conclusion ◮ Each player randomly mutates her state by ◮ dropping her worst idea σi(ℓ−1) ∈ {◭, ◮}ℓ ◮ adding a random idea σ′ ∈ {◭, ◮}ℓ . at chosen intervals, or triggered by bad scores. ◮ This leads to jointly stable populations of players. Positions for stabilization: Minority game Coalgebra of market games D. Pavlovic Games Stable positions Equilibria Positions X hs,idi /A×X A×X hs,idi ̺X X 1 /A×X η=FixX (̺X ◦s) η_ X hs,idi ̺X Coordination /X /X /X O _ ̺X /A×X yield improved social gains E(Σ̺) > m 2 + q. Competition Conclusion Coalgebra of market games Example: Market game (static: X = 1) D. Pavlovic Games For Equilibria ◮ i ∈ n = {0, 1, . . . n − 1} — players (sellers, producers) ◮ Ai = Bi = R — moves, values the payoff A ̺ / B is si − ci ̺ (s) = 0 i if ∀j∈n\{i} . si < sj otherwise Positions Coordination Competition Conclusion Coalgebra of market games Example: Market game (static: X = 1) D. Pavlovic Games For Equilibria ◮ i ∈ n = {0, 1, . . . n − 1} — players (sellers, producers) ◮ Ai = Bi = R — moves, values the payoff A ̺ / B is si − ci ̺ (s) = 0 i if ∀j∈n\{i} . si < sj otherwise where ◮ si is the market price offered by the producer i, ◮ ci is the production cost of i Positions Coordination Competition Conclusion Example: Market game (static: X = 1) Coalgebra of market games D. Pavlovic Games Equilibria Positions Coordination Competition The equilibria 1 s Conclusion / A consist of the strategies si = ci + εi where εi ∈ [pi , qi ] is the desired profit. Example: Market game (with memory and tactics) Coalgebra of market games D. Pavlovic Games Equilibria Marketing tactics (equilibrium selection) Positions Coordination Competition ◮ to win, find ε such that ci + ε < cj + εj for all j , i Conclusion Example: Market game (with memory and tactics) Coalgebra of market games D. Pavlovic Games Equilibria Marketing tactics (equilibrium selection) Positions Coordination Competition ◮ to win, find ε such that ci + ε < cj + εj for all j , i ◮ to profit, maximize among such ε Conclusion Example: Market game (with memory and tactics) Coalgebra of market games D. Pavlovic Games Equilibria Marketing tactics (equilibrium selection) Positions Coordination Competition ◮ to win, find ε such that ci + ε < cj + εj for all j , i ◮ ◮ to profit, maximize among such ε change the game: ◮ sway the buyer to pay more than the lowest price ◮ ◮ lock in, bundling, price discrimination. . . manipulate the market information ◮ advertising, branding. . . Conclusion Stable solution: Second price market game (Static: X = 1) Coalgebra of market games D. Pavlovic Games For Equilibria ◮ i ∈ n = {0, 1, . . . n − 1} — players ◮ Ai = Bi = R — moves, values the payoff A ̺ / B is ⌈si ⌉s − si ̺ (s) = 0 i if ∀j∈n\{i} . si < sj otherwise Positions Coordination Competition Conclusion Stable solution: Second price market game (Static: X = 1) Coalgebra of market games D. Pavlovic Games For Equilibria ◮ i ∈ n = {0, 1, . . . n − 1} — players ◮ Ai = Bi = R — moves, values the payoff A ̺ / B is ⌈si ⌉s − si ̺ (s) = 0 i if ∀j∈n\{i} . si < sj otherwise where ⌈a⌉β = ^ {b ∈ β | a < b} Positions Coordination Competition Conclusion Stable solution: Second price market game (Static: X = 1) Coalgebra of market games D. Pavlovic Games Equilibria The unique equilibrium 1 strategies s / A consists of the Positions Coordination Competition Conclusion si = ci Stable solution: Second price market game (Static, stable, unimplementable) Coalgebra of market games D. Pavlovic Games Equilibria The unique equilibrium 1 strategies s / A consists of the Positions Coordination Competition Conclusion si = ci i.e., ◮ each player announces her production cost ◮ the lowest cost wins the market the profit is ⌈ci ⌉c − ci ◮ ◮ the second lowest cost – the lowest cost Outline Coalgebra of market games D. Pavlovic Games Equilibria Coalgebra of games Positions Conclusion Equilibrium programming Position analysis Conclusion Conclusion: Tasks for coalgebra of games Coalgebra of market games D. Pavlovic Games Equilibria ◮ Positions analyze positions as states in X Conclusion ◮ ◮ coalgebra homomorphisms between games ◮ position bisimilarity construct equilibria as fixed points in A s / A or position-wise X ◮ static 1 ◮ equilibrium at a stationary position 1 s /A hs,xi / A×X