Today s lecture. Competitive Matrix Games. Competitive Matrix Games. Modeling games as hybrid systems. EECE 571M/491M, Spring 2007 Lecture 17

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1 EECE 57M/49M, Spring 007 Lecture 7 Modeling games as hybrid systems oday s lecture Background Matrix games Nash Competitive Equilibrium Nash Bargaining Solution Strategy dynamics: he need for hybrid models moishi@ece.ubc.ca Meeko Oishi, Ph.D. Electrical and Computer Engineering University of British Columbia, BC Hwang, Balakrishnan, and omlin 006) Social dynamics in animal groups wo-player groups hree-player groups Optimal switching policies EECE 57M / 49M Winter 007 Strategies p: frequency Player chooses A q: frequency Player chooses A mixed strategies allowed for both players) Player : Player : u,v ) u,v u,v ) u,v Sum of all strategies for each player must sum to. Players can have more than two strategies Payoffs For a given combination of strategies: Player plays a mixed strategy, p percent A, -p) percent B Player plays a mixed strategy, q percent A, -q) percent B Player receives payoff V p,q) Player receives payoff V p,q) p & u V u & % p) % u u % q) p & v V v & % p) % v v % q) Player : Player : u,v ) u,v u,v ) u,v EECE 57M / 49M Winter 007 EECE 57M / 49M Winter 007 4

2 Nash Competitive Equilibria NCE) u ij ) such that u i j ) u ij ) Points where neither player, considered independently, could do better than the current strategy. NO unique: More than one NCE may exist in a matrix game NCE are Pareto-optimal EECE 57M / 49M Winter 007 and u ij ) u ij ) u ij ) such that there is no u ij, v ij ) u ij ) Player : Player : u,v ) u,v u,v ) u,v 5 Disagreement hreat) Points P) d d % max & p max q Best worst-case benefit each player could receive. Strategy to find d may be different than strategy to find d Without any information about other player s strategy, this is the best one player can accomplish. EECE 57M / 49M Winter 007 min q min p p Uq% p Vq & Player : with abuse of notation, p [p; -p], q [q; -q] here) Player : u,v ) u,v u,v ) u,v 6 wo-player Matrix Games Cooperative Matrix Games Consider the game with EECE 57M / 49M Winter 007 Player : Player :,7) 7, 0,),0) & 7 Strategies x AA : frequency Player and Player jointly choose AA x AB : frequency Player and Player jointly choose AB x BA : frequency Player and Player jointly choose BA x BB : frequency Player and Player jointly choose BB Strategies must sum to A larger region in the value space is now accessible with these strategies than under competitive play Cooperation possible when both players are above their respective threat points Guaranteed to do the same or better than under competition Player : Player : A u,v ) u,v B u,v ) u,v EECE 57M / 49M Winter 007 8

3 Cooperative Matrix Games Payoffs For a given strategy x[x AA, x AB, x BA, x BB ] V x) [ u u u u ]x V x) [ v v v v ]x Player receives payoff V x) Player receives payoff V x) Cooperative Matrix Games Nash Bargaining Solution NBS) V NBS max V x) d x ) V x) d ) Point for which both players can optimize their joint benefit NBS solution can be less than what one player might receive from an NCE solution Implied notion -- repeated interactions for which threats are credible and all information known EECE 57M / 49M Winter 007 Player : Player : u,v ) u,v u,v ) u,v 9 EECE 57M / 49M Winter 007 Player : Player : u,v ) u,v u,v ) u,v 0 Motivation for Hybrid Systems Competitive Dynamic Games Strategies are continuous variables, evolving dynamically In some settings e.g., animal groups) games are occurring continuously Players update their strategies according to their current benefit Players move in directions of increased benefit Dynamics differ when players are competing or cooperating Based on dynamics for gamete allocation EECE 57M / 49M Winter 007 Dynamic strategies As before, p, q are percentage of time spent playing strategy A Dependent on individual benefit gradients Payoffs As defined previously p & u V u & % p) % u u % q) p & v V v & % p) % v v % q) EECE 57M / 49M Winter 007 Player : p p) V p p,q) V p,q) p q q) V q p,q) V p,q) q Player : u,v ) u,v u,v ) u,v

4 Cooperative Dynamic Games Dynamic strategies As before, x[xaa, xab, xba, xbb] is percentage of time spent playing strategies AA, AB, BA, BB, respectively Dependent on joint benefit gradient x ij Dynamic Games x ij x ij ) Vx) d)v x) d ) V x) d )V x) d )) x Payoffs V x) [ u u u u ] x V x) [v v v v ] x Player : A B Stability in the sense of Lyapunov Competitive dynamics converge locally to the NCE Cooperative dynamics converge to the NBS for any x which result in V,V) above the threat point Differential algebraic equations due to fact that strategies must sum to. Polynomial dynamics, polynomial output payoff) Player : A B u,v ) u,v ) % u,v ) u,v )& EECE 57M / 49M Winter 007 EECE 57M / 49M Winter wo-player hybrid game Social dynamics in animals Cooperation Competition Competition Darwin, 86, Sexual selection theory Cooperation occurs when both players benefits are above their threat point J. Nash, 950, NCE Maynard Smith, 98. Evolutionary stable strategies Competition occurs when either player s benefit is below their threat point 5 Bluegill Sunfish Optimization of individual fitness J. Nash, 95, NBS Roughgarden, 006, Social selection theory EECE 57M / 49M Winter 007 Peacock-Wrasse Cooperation d hreat Point Competition for genes Well-documented variations in nature Oystercatcher Optimization of joint fitness EECE 57M / 49M Winter 007 Pukeko 6

5 Peacock-wrasse Bluegill Sunfish Female : Evidence for cooperative nesting Sea Floor Nest strategies Search,) 0,0 Male : Stay 0,0),)& Why does the nesting male sometimes allow the satellite male to fertilize eggs in the Defend nest? Male : Allow EECE 57M / 49M Winter hree-player hybrid games Each player s strategy evolves continuously opology indicates alliances, which are mutual Different dynamics depending on level of competition or cooperation Rational dynamics Minimum set of conditions for mode to be feasible e.g., based on threat points) Competition between all three players Each player has strategies vs {} 8 hree-player hybrid games EECE 57M / 49M Winter 007 vs vs EECE 57M / 49M Winter 007 Male : Avoid Intrude,0) 0,) %,0) 0.5,0.5 vs {} vs {} Cooperation between all three players otal of 8 strategies {} 9 EECE 57M / 49M Winter 007 0

6 hree-player hybrid games hree-player hybrid games,{} x f a x) y V a x),, x f 0 x) y V 0 x),{} x f b x) y V b x) {} x f d x) y V d x),{} x f c x) y V c x) )) Competition,,): x ijk x x ) V x) d ijk ijk m m m Cooperation {}): V mx) d m ) x ijk p p) V m p p,q,r) V p,q,r) p q q) V x Coalition,{}): x ij ij x ij ) V x,r)v x,r)) q p,q,r) V V x,r)v x,r) x ij p,q,r) q r r) V r r) V r x,r) r p,q,r) V EECE V p,q,r) 57M / 49M r Winter 007 x,r) r Divide the Dollar,{} EECE 57M / 49M Winter 007 Oystercatchers Monogamous pairs, polygynous trios rios: 50% shared nests, 50% separate nests How can hybrid dynamical games explain this phenomena? Male : A Female : Aggress Bond Female : Aggress,,),0,) % Bond,,),,4 Male : B F : F: A,,),,) % B 0,,),,4 Switching coalitions Optimal switching policies Dynamic programming with multiple objectives and state discontinuities Pursuer/evader aircraft games Comparison to standard game theory results bargaining sets, core, etc.) Analysis of hybrid switching Predicting switching coalitions periodic orbits) EECE 57M / 49M Winter 007 EECE 57M / 49M Winter 007 4

7 Oystercatchers Summary Coalition dynamics {} Matrix games competitive and cooperative) Hybrid dynamical games: social dynamics in animal groups Modeling Stability within each coalition through Lyapunov functions Relationship to traditional discrete) models,,,{} Predicting ecological phenomenon Oystercatcher trios polygynous), 50% aggressive, 50% cooperative Bluegill sunfish mating strategies Peacock-Wrasse mating strategies Optimal coalition formation and dissolution EECE 57M / 49M Winter EECE 57M / 49M Winter 007 6