Equilibrium computation: Part 1
|
|
- Ursula Preston
- 9 years ago
- Views:
Transcription
1 Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
2 Outline 1 Models and solution concepts Mechanisms in strategic form Solution concepts 2 Non equilibrium solution concept computation Finding dominated actions Finding never best response actions 3 Computing a Nash equilibrium with strategic form games Matrix games Bimatrix games Polymatrix games 4 Computing correlation based equilibria with strategic form games Computing a correlated equilibrium Computing a leader follower equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
3 Game model Definition A game is formally defined by a pair: Mechanism M, defining the rules of the game Strategiesσ, defining the behavior of each agent in the game Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
4 Game model Definition A game is formally defined by a pair: Mechanism M, defining the rules of the game Strategiesσ, defining the behavior of each agent in the game Mechanisms There are three main classes of mechanisms: Strategic form mechanisms: agents play without observing the actions undertaken by the opponents (simultaneous games) Extensive form mechanisms: there is a sequential tree based structure according which an agent can observe some opponents actions Stochastic form mechanisms: there is a sequential graph based structure according which an agent can observe some opponents actions Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
5 Games in strategic form (1) Definition A strategic form mechanism is a tuple M = (N,{A} i N, X, f,{u} i N ) N: set of agents A i : set of actions available to agent i X: set of outcomes f : i N A i X: outcome function U i : X R: utility function of agent i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
6 Games in strategic form (2) Example: Rock Paper Scissors N = {agent 1, agent 2} A 1 = A 2 = {R, P, S} X = {win1, win2, tie} f(r, S) = f(p, R) = f(s, P) = win 1, f(s, R) = f(r, P) = f(p, S) = win2, tie otherwise U i (wini) = 1, U i (win i) = 1, U i (tie) = 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
7 Games in strategic form (2) Example: Rock Paper Scissors N = {agent 1, agent 2} A 1 = A 2 = {R, P, S} X = {win1, win2, tie} f(r, S) = f(p, R) = f(s, P) = win 1, f(s, R) = f(r, P) = f(p, S) = win2, tie otherwise U i (wini) = 1, U i (win i) = 1, U i (tie) = 0 Matrix based representation agent 1 agent 2 R P S R 0, 0 1, 1 1, 1 P 1, 1 0, 0 1, 1 S 1, 1 1, 1 0, 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
8 Games in strategic form (3) Example: three player game A 1 = {a, b} A 2 = {L, R} A 3 = {A, B, C} L R a 2, 2, 1 0, 3, 0 b 3, 0, 2 1, 1, 4 A L R a 2, 3, 0 0, 4, 1 b 3, 1, 2 1, 2, 0 B L R a 2, 1, 0 1, 0, 2 b 0, 3, 1 2, 3, 1 C Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
9 Matrix based games Classification Matrix game: the agents utilities can be represented by a unique matrix (this happens with two agent constant sum games: U 1 + U 2 = constant for every entry) Bimatrix game: two agent general sum games Polymatrix game: the utility U i of each agent i can be expressed as a set of matrices U i,j depending only on the actions of agent i and agent j with non polymatrix games, U i has j N A j entries with polymatrix games, U i has A i j N,j i A j entries Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
10 Strategies Definition A strategy σ i of agent i is a probability distribution over the actions A i Call x i,j the probability with which agent i plays action j and x i the vector of x i,j, we need that x i 0 1 T x i = 1 A strategy profileσ is the collection of one strategy per agent, σ = (σ 1,...,σ N ) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
11 Strategies Definition A strategy σ i of agent i is a probability distribution over the actions A i Call x i,j the probability with which agent i plays action j and x i the vector of x i,j, we need that x i 0 1 T x i = 1 A strategy profileσ is the collection of one strategy per agent, σ = (σ 1,...,σ N ) Example With Rock Paper Scissors games can be: x 1 = x 1,R = 0.2 x 1,P = 0.8 x 2 = x 2,R = 0.6 x 2,P = 0.0 x 1,S = 0.0 x 2,S = 0.4 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
12 Expected utility (1) Definition The expected utility of an agent i related to an action j is: U i x k k N,k i j where(a) j is the j th row of matrix A U i k N,k i x k is the vector of expected utilities of agent i The expected utility of an agent i related to a strategy x i is: x T i U i k N,k i x k Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
13 Expected utility (2) Example U 1 = x 1 = x 2 = The expected utilities related to each action of agent 1 are: = The expected utility related to the strategy of agent 1 is: [ ] = Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
14 Game equivalence Definition Given two games with utility functions U 1,...,U N and U 1,...,U N respectively, if, for every i N, there is an affine transformation between U i and U i such that U i = α iu i +β i A 1 where A 1 is a matrix of ones, then the two games are equivalent Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
15 Game equivalence Definition Given two games with utility functions U 1,...,U N and U 1,...,U N respectively, if, for every i N, there is an affine transformation between U i and U i such that U i = α iu i +β i A 1 where A 1 is a matrix of ones, then the two games are equivalent Example U 1 = U 1 = = Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
16 Solutions and solution concepts Definition Given: The strategy x i of each agent i The beliefˆx i j each agent i has over the strategy x j of agent j A solution is a pair(σ,µ), whereµis the set of agents beliefs, such that Rationality constraints: the strategies of each agent are optimal w.r.t. the beliefs Information constraints: the beliefs of each agent are somehow consistent w.r.t. the opponents strategies Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
17 Solutions and solution concepts Definition Given: The strategy x i of each agent i The beliefˆx i j each agent i has over the strategy x j of agent j A solution is a pair(σ,µ), whereµis the set of agents beliefs, such that Rationality constraints: the strategies of each agent are optimal w.r.t. the beliefs Information constraints: the beliefs of each agent are somehow consistent w.r.t. the opponents strategies Definition A solution concept defines the set of rationality and information constraints Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
18 Solution concept classification Non equilibrium solution concepts Dominance and iterated dominance Never best response and iterated never best response Maxmin strategy and minmax strategy Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
19 Solution concept classification Non equilibrium solution concepts Dominance and iterated dominance Never best response and iterated never best response Maxmin strategy and minmax strategy Equilibrium solution concepts without correlation Nash relaxations: conjectural equilibrium, self confirming equilibrium Nash Nash refinements: perfect equilibrium, proper equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
20 Solution concept classification Non equilibrium solution concepts Dominance and iterated dominance Never best response and iterated never best response Maxmin strategy and minmax strategy Equilibrium solution concepts without correlation Nash relaxations: conjectural equilibrium, self confirming equilibrium Nash Nash refinements: perfect equilibrium, proper equilibrium Equilibrium solution concepts with correlation One agent based correlation: leader follower/stackelberg/committment equilibrium Device based correlation: correlated equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
21 Dominance (1) Definition Action j A i is strictly dominated if there is a strategy x over A that, for every action of the opponents, provides an expected utility larger than action j e T j U i < x T U i where e j is a vector of zeros except for position j wherein there is 1 Example agent 1 Action C is dominated by action B agent 2 D E F A 4, 1 1, 2 1, 3 B 1, 4 4, 0 4, 1 C 0, 1 2, 5 2, 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
22 Dominance (2) Weakly dominance Action j A i is weakly dominated if there is a strategy x over A that, for every action of the opponents, provides an expected utility equal to or larger than action j Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
23 Dominance (2) Weakly dominance Action j A i is weakly dominated if there is a strategy x over A that, for every action of the opponents, provides an expected utility equal to or larger than action j Dominance and rationality No rational agent will play an action that is strictly dominated Strictly dominated actions can be safely removed from the game, never being played The application of strong dominance leads to a reduced game that is equivalent to the original one Weakly dominated actions could be played by agents Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
24 Dominance and mixed strategies Property Dominance with mixed strategies is stronger than with pure strategies Example agent 1 agent 2 D E F A 4, 1 1, 2 1, 3 B 1, 4 4, 0 4, 1 C 2, 1 2, 5 2, 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
25 Dominance and mixed strategies Property Dominance with mixed strategies is stronger than with pure strategies Example agent 1 Dominance in pure strategies agent 2 D E F A 4, 1 1, 2 1, 3 B 1, 4 4, 0 4, 1 C 2, 1 2, 5 2, 0 No action of the agent 1 is dominated by another action Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
26 Dominance and mixed strategies Property Dominance with mixed strategies is stronger than with pure strategies Example agent 1 Dominance in pure strategies agent 2 D E F A 4, 1 1, 2 1, 3 B 1, 4 4, 0 4, 1 C 2, 1 2, 5 2, 0 No action of the agent 1 is dominated by another action Dominance in mixed strategies Action C is dominated by x = [ ] Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
27 Dominance with more than two agents Example L R a 2, 2, 1 0, 3, 0 b 3, 0, 2 1, 1, 4 A L R a 2, 3, 0 0, 4, 1 b 3, 1, 2 1, 2, 0 B L R a 2, 1, 0 1, 0, 2 b 3, 3, 1 2, 3, 1 C Action a is dominated by action b Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
28 Dominance as a solution concept Comments Dominance does not require any assumption over the information available to each agent except for the knowledge of own utility Dominance prescribes what actions are to play and what are not to play independently of the opponents strategies Dominance does not prescribe any strategy over the non dominated actions We have an equilibrium in dominant strategies if dominance removes all the actions except one for every agent Example agent 2 S C agent 1 S 2, 2 0, 3 C 3, 0 1, 1 agent 1 agent 2 H T H 2, 0 0, 2 T 0, 2 2, 0 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
29 Iterated dominance Definition Under the assumption of complete information over the utility and common information over rationality and utilities, each agent can forecast the dominated actions of the opponents and iteratively remove her own actions Example agent 1 agent 2 D E F A 3, 2 2, 1 2, 0 B 0, 2 0, 5 3, 3 C 0, 1 1, 2 1, 4 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
30 Best response Definition The best response of agent i is an action that maximizes her expected utility given the strategies of the opponents as input BR i (σ i ) = arg max j A i et j U i x k where x k are given k N,k=i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
31 Best response Definition The best response of agent i is an action that maximizes her expected utility given the strategies of the opponents as input BR i (σ i ) = arg max j A i et j U i x k where x k are given Comments BR i (σ i ) can return multiple actions k N,k=i A rational agent will play only best response actions Any mixed strategy over best response actions is a best response Any non never best response action is said rationalizable Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
32 Never best response Definition A never best response of agent i is an action j such that there is not any opponents strategy profile such that action j is a best response j BR i (σ i ) σ i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
33 Never best response Definition A never best response of agent i is an action j such that there is not any opponents strategy profile such that action j is a best response j BR i (σ i ) σ i Comments No rational agent will play never best response actions Never best response actions can be safely removed Rationalizability requires each agent to know her own utilities, no assumption is required over the information on the opponents utilities and rationality Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
34 Never best response Definition A never best response of agent i is an action j such that there is not any opponents strategy profile such that action j is a best response j BR i (σ i ) σ i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
35 Never best response Definition A never best response of agent i is an action j such that there is not any opponents strategy profile such that action j is a best response j BR i (σ i ) σ i Comments No rational agent will play never best response actions Never best response actions can be safely removed When information on the utilities and rationality is complete and common, rationalizability can be iterated Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
36 Rationalizability and dominance (1) Comments Dominance and rationalizability are equivalent with two agents (the proof is by strong duality) With more than two agents, every dominated action is a never best response, but the reverse may not hold (rationalizability removes a larger number of actions than dominance) The main difference: Dominance is similar to rationalizability, but it implicitly assumes that the opponents correlate their strategy as a unique agent Rationalizability explicitly considers each opponent as a different uncorrelated agent If an action is dominated when the opponents can correlate is also dominated when they cannot If an action is dominated when the opponents cannot correlate, it may be not when they can Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
37 Rationalizability and dominance (2) Example L R a 0, 0, 0 0, 0, 0 b 8, 8, 8 0, 0, 0 L R a 0, 0, 0 8, 8, 8 b 0, 0, 0 0, 0, 0 L R a 4, 4, 4 0, 0, 0 b 0, 0, 0 4, 4, 4 L R a 3, 3, 3 3, 3, 3 b 3, 3, 3 3, 3, 3 A B C D Action D is not strictly dominated, but it is a never best response Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
38 Maxmin Assumptions An agent does not know anything about her opponents An agent aims at maximize her utility in the worst case (safety level) Definition A maxmin strategyσ of agent i is defined as: σ = arg max mine[u i ] σ i σ i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
39 Minmax Assumptions An agent knows the utility of the opponent An agent aims at minimize the opponent expected utility Definition A minmax strategyσ of agent i is defined as: σ = arg min maxe[u i ] σ i σ i Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
40 Nash equilibrium (1) Assumptions Agents do not communicate before playing Agents know the utilities of the opponents and this information is common Definition A Nash equilibrium is a strategy profile(x 1,...,x n) such that: (x i )T U i j N,j i x j x T i U i j N,j i x j x i, i N Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
41 Nash equilibrium (1) Assumptions Agents do not communicate before playing Agents know the utilities of the opponents and this information is common Definition A Nash equilibrium is a strategy profile(x 1,...,x n) such that: Comments (x i )T U i j N,j i x j x T i U i j N,j i x j x i, i N In a Nash equilibrium, no agent can more by changing her strategy given that the opponents do not change (i..e, every x i is a randomization over best responses) Coalition deviations are not considered Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
42 Nash equilibrium (2) Definition A Nash equilibrium is a strategy profile(x 1,...,x n ) such that: (x i )T U i e T k U i k A i, i N Comments j N,j i x j j N,j i We can substitute x i (infinite constraints) with k A i ( A i constraints) because x T i U i j N,j i x j is a convex combination of different e T k U i j N,j i x j x T i U i j N,j i x j is smaller than or equal to max k e T k U i j N,j i x j since we cannot know what is k with the largest e T k U i j N,j i x j, we impose to be larger than equal to all the e T k U i j N,j i x j We obtain a finite number of constraints that is linear in the size of the game x j Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
43 Nash theorem Theorem Every finite game admits at least a Nash equilibrium in mixed strategies Comments The proof is by Brouwer fixed point theorem: a Nash equilibrium is a fixed point Pure strategies Nash equilibria may not exist (e.g., Matching penny) Multiple equilibria can coexist With continuous games, the things are more complicated (a continuous game may not admit any Nash equilibrium, neither in mixed strategies) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
44 Example (1) Pure strategy equilibrium agent 1 Pure strategy equilibrium agent 1 agent 2 D E F A 1, 3 2, 1 1, 0 B 3, 2 0, 5 2, 3 C 0, 1 1, 2 3, 3 agent 2 D E F A 6, 2 2, 1 1, 6 B 3, 2 3, 3 2, 3 C 0, 6 1, 2 3, 3 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
45 Example (2) Multiple pure strategy equilibria agent 1 No pure strategy equilibrium agent 1 agent 2 D E F A 6, 2 2, 1 1, 6 B 3, 2 3, 3 2, 3 C 0, 6 1, 2 9, 9 agent 2 D E F A 6, 2 2, 1 1, 6 B 3, 2 0, 3 2, 3 C 0, 6 1, 2 3, 3 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
46 Nash equilibrium and Pareto efficiency Example agent 2 S C agent 1 S 2, 2 0, 3 C 3, 0 1, 1 There is a unique Nash equilibrium(c, C) (C, C) is Pareto dominated by (S, S) (C, C) is the unique Pareto dominated strategy profile There is no relationship between Pareto dominance and Nash equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
47 Perturbed games (1) Perturbation Given a set of action A i, a perturbation over it corresponds to a probability function f i,j with j A i and j A i f i,j < 1 Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
48 Perturbed games (1) Perturbation Given a set of action A i, a perturbation over it corresponds to a probability function f i,j with j A i and j A i f i,j < 1 Parametric perturbation Perturbation f i,j = f i,j (ǫ) withǫ [0, 1] Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
49 Perturbed games (1) Perturbation Given a set of action A i, a perturbation over it corresponds to a probability function f i,j with j A i and j A i f i,j < 1 Parametric perturbation Perturbation f i,j = f i,j (ǫ) withǫ [0, 1] Perturbed game Given a perturbation f i,j, a perturbed game is a game in which strategies are constrained as: i N, j A i : x i,j f i,j Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
50 Perturbed games (1) Perturbation Given a set of action A i, a perturbation over it corresponds to a probability function f i,j with j A i and j A i f i,j < 1 Parametric perturbation Perturbation f i,j = f i,j (ǫ) withǫ [0, 1] Perturbed game Given a perturbation f i,j, a perturbed game is a game in which strategies are constrained as: i N, j A i : x i,j f i,j Perturbation and Nash equilibrium The introduction of perturbation (i.e., a perturbation game) affects the set of Nash equilibria Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
51 Perturbed games (2) Example agent 1 agent 2 C D A 10, 10 0, 0 B 0, 0 1, 1 Perturbation: f 1,A = 0.2, f 1,B = 0.2, f 2,C = 0.2, f 2,D = 0.2 (A, C) and (B, D) are Nash equilibria without perturbation (0.8A+0.2B, 0.8C+ 0.2D) is a Nash equilibrium with perturbation: all the probability except for the perturbation is put on(a, C) (0.2A+0.8B, 0.2C+ 0.BD) is not a Nash equilibrium with perturbation: all the probability except for the perturbation cannot put on(b, D) Perturbation: f 1,A = 0.05, f 1,B = 0.05, f 2,C = 0.05, f 2,D = 0.05 (B, D) is a Nash equilibrium without perturbation (0.2A+0.8B, 0.2C+ 0.BD) is a Nash equilibrium with perturbation: all the probability except for the perturbation is put on(b, D) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
52 Perfect equilibrium (1) Definition A strategy profileσ is a perfect equilibrium if there is a f i,j (ǫ) such that, called σ (ǫ) a sequence of Nash equilibria for any ǫ [0,ǫ 0 ] of the associated perturbed games,σ (ǫ) σ as ǫ 0 Example agent 1 agent 2 C D A 1, 1 0, 0 B 0, 0 0, 0 For every f 1,A (ǫ) > 0, action D is not a best response For every f 2,C (ǫ) > 0, action B is not a best response (B, D) is a Nash equilibrium, but it is not perfect (A, C) is a perfect equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
53 Perfect equilibrium (2) Properties An equilibrium is perfect if it keeps to be a Nash equilibrium when minimally perturbed Every finite game admits at least a perfect equilibrium Every perfect equilibrium is a Nash equilibrium in which no weakly dominated action is played The vice versa (i.e., every Nash equilibrium in which no weakly dominated action is played is a perfect equilibrium) is true only for two player games There is not relationship between perfect equilibrium and Pareto efficiency We can safely consider only f i,j (ǫ) that are polynomial inǫ Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
54 Perfect equilibrium (3) Example C D A 1, 1, 1 1, 0, 1 B 1, 1, 1 0, 0, 1 E C D A 1, 1, 1 0, 0, 0 B 0, 1, 0 1, 0, 0 F F is weakly dominated for agent 3 D is weakly dominated for agent 2 (A, C, E) and (B, C, E) are Nash equilibria without weakly dominated actions (A, C, E) is not perfect Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
55 Perfect equilibrium (3) Example agent 1 agent 2 C D A 1, 1 10, 0 B 0, 10 10, 10 There are two pure strategy Nash equilibria (A, C) and(b, D) Actions B and D are weakly dominated The unique perfect Nash is (A, C) (B, D) Pareto dominates (A, C) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
56 Perfect equilibrium (5) Example agent 1 agent 2 A B agent 1 a 1, 1 0, 0 b 0, 0 0, 0 agent 2 A B C a 1, 1 0, 0-1,2 b 0, 0 0, 0 0,-2 c 2, 1 2, 0-2,-2 Without c and C, the unique perfect equilibrium is (a, A) With c and C, (b, B) is a perfect equilibrium The introduction of strictly dominated actions may change the set of perfect equilibria Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
57 Proper equilibrium (1) Perfection weakness The perfect equilibrium is sensible to weakly dominated actions Aim The design of a solution concept refining Nash equilibrium that is not sensible to weakly dominated actions Properness idea A proper equilibrium is a perfect equilibrium with a specific perturbation: given two actions j and k of agent i, if j provides a utility strictly larger than k, then perturbation f i,k is subject to f i,k ǫf i,j In other words The perturbation has the property that a good action must be played (due to perturbation) with probability larger than the probability of a bad action Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
58 Proper equilibrium (2) Properties Every game admits at least a proper equilibrium The proper equilibrium removes weakly dominated strategies with two player games With more agents, the proper equilibrium may not remove weakly dominated strategies Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
59 Correlated equilibrium (1) Assumptions Agents can correlate in some way Typically, a correlation device is considered that sends different signals to each agent Definition A correlated equilibrium is a tuple(v,π,σ), where v is a tuple of random variables v = (v 1,...,v n ) with respective domains D = (D 1,...,D n ),π is a joint distribution over v,σ = (σ 1,...,σ n ) is a vector of mappings σ i : D i A i, and for each agent i and every mapping σ i is the case that: d Dπ(d i, d i )U i (σ i (d i ),σ i (d i )) d D It is possible to limit strategies σ i to be pure π(d i, d i )U i (σ i (d i),σ i (d i )) Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
60 Correlated equilibrium (2) Properties Every Nash equilibrium is a correlated equilibrium in which there is only one signal per agent A correlated equilibrium may be not a Nash equilibrium Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
61 Leader follower equilibrium (1) Assumptions An agent, called leader, can announce (commit to) her strategy to the opponents The other agents, called followers, act knowing the commitment The announce must be credible Definition A leader follower equilibrium is a strategy profile in which the expected utility of the leader is maximized given that the followers act knowing the strategy of the leader Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium Duke University, computation: USA Part ) / 120
4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More information1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.
Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationLECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method
LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right
More informationCOMPUTING EQUILIBRIA FOR TWO-PERSON GAMES
COMPUTING EQUILIBRIA FOR TWO-PERSON GAMES Appeared as Chapter 45, Handbook of Game Theory with Economic Applications, Vol. 3 (2002), eds. R. J. Aumann and S. Hart, Elsevier, Amsterdam, pages 1723 1759.
More informationLinear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.
Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.
More informationLinear Programming: Chapter 11 Game Theory
Linear Programming: Chapter 11 Game Theory Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/ rvdb Rock-Paper-Scissors
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More information3. Linear Programming and Polyhedral Combinatorics
Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the
More informationGame Theory: Supermodular Games 1
Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
More information6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games
6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitely-repeated prisoner s dilemma
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationI d Rather Stay Stupid: The Advantage of Having Low Utility
I d Rather Stay Stupid: The Advantage of Having Low Utility Lior Seeman Department of Computer Science Cornell University lseeman@cs.cornell.edu Abstract Motivated by cost of computation in game theory,
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationImperfect monitoring in communication networks
Journal of Economic Theory (00) www.elsevier.com/locate/jet Imperfect monitoring in communication networks Michael McBride University of California, Irvine, Social Science Plaza, Irvine, CA -00, USA Received
More informationDiscrete Optimization
Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using
More informationThe Max-Distance Network Creation Game on General Host Graphs
The Max-Distance Network Creation Game on General Host Graphs 13 Luglio 2012 Introduction Network Creation Games are games that model the formation of large-scale networks governed by autonomous agents.
More informationMicroeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012. (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3
Microeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012 1. Subgame Perfect Equilibrium and Dominance (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3 Highlighting
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationDuality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725
Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T
More informationComputational Learning Theory Spring Semester, 2003/4. Lecture 1: March 2
Computational Learning Theory Spring Semester, 2003/4 Lecture 1: March 2 Lecturer: Yishay Mansour Scribe: Gur Yaari, Idan Szpektor 1.1 Introduction Several fields in computer science and economics are
More informationHow to Solve Strategic Games? Dominant Strategies
How to Solve Strategic Games? There are three main concepts to solve strategic games: 1. Dominant Strategies & Dominant Strategy Equilibrium 2. Dominated Strategies & Iterative Elimination of Dominated
More informationA Game Theoretical Framework on Intrusion Detection in Heterogeneous Networks Lin Chen, Member, IEEE, and Jean Leneutre
IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL 4, NO 2, JUNE 2009 165 A Game Theoretical Framework on Intrusion Detection in Heterogeneous Networks Lin Chen, Member, IEEE, and Jean Leneutre
More informationCan linear programs solve NP-hard problems?
Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf Linear programs Can linear programs solve NP-hard problems? p. 2/9 Can linear programs solve
More informationGame Theory 1. Introduction
Game Theory 1. Introduction Dmitry Potapov CERN What is Game Theory? Game theory is about interactions among agents that are self-interested I ll use agent and player synonymously Self-interested: Each
More informationCompact Representations and Approximations for Compuation in Games
Compact Representations and Approximations for Compuation in Games Kevin Swersky April 23, 2008 Abstract Compact representations have recently been developed as a way of both encoding the strategic interactions
More information1 Nonzero sum games and Nash equilibria
princeton univ. F 14 cos 521: Advanced Algorithm Design Lecture 19: Equilibria and algorithms Lecturer: Sanjeev Arora Scribe: Economic and game-theoretic reasoning specifically, how agents respond to economic
More informationManipulability of the Price Mechanism for Data Centers
Manipulability of the Price Mechanism for Data Centers Greg Bodwin 1, Eric Friedman 2,3,4, and Scott Shenker 3,4 1 Department of Computer Science, Tufts University, Medford, Massachusetts 02155 2 School
More informationA Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem
A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,
More informationTransportation Polytopes: a Twenty year Update
Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,
More informationSensitivity analysis of utility based prices and risk-tolerance wealth processes
Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationRecovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach
MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical
More informationThe effect of exchange rates on (Statistical) decisions. Teddy Seidenfeld Mark J. Schervish Joseph B. (Jay) Kadane Carnegie Mellon University
The effect of exchange rates on (Statistical) decisions Philosophy of Science, 80 (2013): 504-532 Teddy Seidenfeld Mark J. Schervish Joseph B. (Jay) Kadane Carnegie Mellon University 1 Part 1: What do
More informationCollaboration for Truckload Carriers
Submitted to Transportation Science manuscript (Please, provide the mansucript number!) Collaboration for Truckload Carriers Okan Örsan Özener H. Milton Stewart School of Industrial and Systems Engineering,
More informationBargaining Solutions in a Social Network
Bargaining Solutions in a Social Network Tanmoy Chakraborty and Michael Kearns Department of Computer and Information Science University of Pennsylvania Abstract. We study the concept of bargaining solutions,
More informationNonlinear Optimization: Algorithms 3: Interior-point methods
Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,
More informationSimplex method summary
Simplex method summary Problem: optimize a linear objective, subject to linear constraints 1. Step 1: Convert to standard form: variables on right-hand side, positive constant on left slack variables for
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationThe Multiplicative Weights Update method
Chapter 2 The Multiplicative Weights Update method The Multiplicative Weights method is a simple idea which has been repeatedly discovered in fields as diverse as Machine Learning, Optimization, and Game
More informationIntroduction to Support Vector Machines. Colin Campbell, Bristol University
Introduction to Support Vector Machines Colin Campbell, Bristol University 1 Outline of talk. Part 1. An Introduction to SVMs 1.1. SVMs for binary classification. 1.2. Soft margins and multi-class classification.
More information56:171 Operations Research Midterm Exam Solutions Fall 2001
56:171 Operations Research Midterm Exam Solutions Fall 2001 True/False: Indicate by "+" or "o" whether each statement is "true" or "false", respectively: o_ 1. If a primal LP constraint is slack at the
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More informationDuality in Linear Programming
Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow
More informationA Game Theoretic Formulation of the Service Provisioning Problem in Cloud Systems
A Game Theoretic Formulation of the Service Provisioning Problem in Cloud Systems Danilo Ardagna 1, Barbara Panicucci 1, Mauro Passacantando 2 1 Politecnico di Milano,, Italy 2 Università di Pisa, Dipartimento
More informationOligopoly: Cournot/Bertrand/Stackelberg
Outline Alternative Market Models Wirtschaftswissenschaften Humboldt Universität zu Berlin March 5, 2006 Outline 1 Introduction Introduction Alternative Market Models 2 Game, Reaction Functions, Solution
More information! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one
More informationBilevel Models of Transmission Line and Generating Unit Maintenance Scheduling
Bilevel Models of Transmission Line and Generating Unit Maintenance Scheduling Hrvoje Pandžić July 3, 2012 Contents 1. Introduction 2. Transmission Line Maintenance Scheduling 3. Generating Unit Maintenance
More informationTwo-Stage Stochastic Linear Programs
Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic
More informationChapter 11 Monte Carlo Simulation
Chapter 11 Monte Carlo Simulation 11.1 Introduction The basic idea of simulation is to build an experimental device, or simulator, that will act like (simulate) the system of interest in certain important
More informationIEOR 4404 Homework #2 Intro OR: Deterministic Models February 14, 2011 Prof. Jay Sethuraman Page 1 of 5. Homework #2
IEOR 4404 Homework # Intro OR: Deterministic Models February 14, 011 Prof. Jay Sethuraman Page 1 of 5 Homework #.1 (a) What is the optimal solution of this problem? Let us consider that x 1, x and x 3
More informationINTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models
Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is
More information! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three
More informationIntegrating Benders decomposition within Constraint Programming
Integrating Benders decomposition within Constraint Programming Hadrien Cambazard, Narendra Jussien email: {hcambaza,jussien}@emn.fr École des Mines de Nantes, LINA CNRS FRE 2729 4 rue Alfred Kastler BP
More informationMeasuring the Performance of an Agent
25 Measuring the Performance of an Agent The rational agent that we are aiming at should be successful in the task it is performing To assess the success we need to have a performance measure What is rational
More informationUCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory
UCLA Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.
More informationChapter 7. Sealed-bid Auctions
Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationDiscuss the size of the instance for the minimum spanning tree problem.
3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can
More informationScheduling Shop Scheduling. Tim Nieberg
Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no
More information3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max
SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,
More informationMinimizing costs for transport buyers using integer programming and column generation. Eser Esirgen
MASTER STHESIS Minimizing costs for transport buyers using integer programming and column generation Eser Esirgen DepartmentofMathematicalSciences CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG
More informationAdaptive Linear Programming Decoding
Adaptive Linear Programming Decoding Mohammad H. Taghavi and Paul H. Siegel ECE Department, University of California, San Diego Email: (mtaghavi, psiegel)@ucsd.edu ISIT 2006, Seattle, USA, July 9 14, 2006
More informationSome representability and duality results for convex mixed-integer programs.
Some representability and duality results for convex mixed-integer programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012. Introduction About Motivation Mixed integer
More informationBig Data - Lecture 1 Optimization reminders
Big Data - Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Big Data - Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Schedule Introduction Major issues Examples Mathematics
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More information1 Representation of Games. Kerschbamer: Commitment and Information in Games
1 epresentation of Games Kerschbamer: Commitment and Information in Games Game-Theoretic Description of Interactive Decision Situations This lecture deals with the process of translating an informal description
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationHow To Solve A Minimum Set Covering Problem (Mcp)
Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix
More informationWhen is Reputation Bad? 1
When is Reputation Bad? 1 Jeffrey Ely Drew Fudenberg David K Levine 2 First Version: April 22, 2002 This Version: November 20, 2005 Abstract: In traditional reputation theory, the ability to build a reputation
More informationSocial Media Mining. Network Measures
Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users
More informationNotes from Week 1: Algorithms for sequential prediction
CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 22-26 Jan 2007 1 Introduction In this course we will be looking
More informationA Constraint Programming based Column Generation Approach to Nurse Rostering Problems
Abstract A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Fang He and Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) Group School of Computer Science,
More informationWeek 7 - Game Theory and Industrial Organisation
Week 7 - Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number
More informationOptimization Modeling for Mining Engineers
Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2
More informationEcon 430 Lecture 9: Games on Networks
Alper Duman Izmir University Economics, May 10, 2013 Semi-Anonymous Graphical Games refer to games on networks in which agents observe what their neighbours do The number of agents taking a specific action
More informationPermutation Betting Markets: Singleton Betting with Extra Information
Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu
More informationGames Manipulators Play
Games Manipulators Play Umberto Grandi Department of Mathematics University of Padova 23 January 2014 [Joint work with Edith Elkind, Francesca Rossi and Arkadii Slinko] Gibbard-Satterthwaite Theorem All
More informationLinear Programming. April 12, 2005
Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first
More informationComputational Game Theory and Clustering
Computational Game Theory and Clustering Martin Hoefer mhoefer@mpi-inf.mpg.de 1 Computational Game Theory? 2 Complexity and Computation of Equilibrium 3 Bounding Inefficiencies 4 Conclusion Computational
More informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationNP-Hardness Results Related to PPAD
NP-Hardness Results Related to PPAD Chuangyin Dang Dept. of Manufacturing Engineering & Engineering Management City University of Hong Kong Kowloon, Hong Kong SAR, China E-Mail: mecdang@cityu.edu.hk Yinyu
More informationA Game Theoretical Framework for Adversarial Learning
A Game Theoretical Framework for Adversarial Learning Murat Kantarcioglu University of Texas at Dallas Richardson, TX 75083, USA muratk@utdallas Chris Clifton Purdue University West Lafayette, IN 47907,
More informationFactoring Games to Isolate Strategic Interactions
Factoring Games to Isolate Strategic Interactions George B. Davis, Michael Benisch, Kathleen M. Carley, and Norman M. Sadeh January 2007 CMU-ISRI-06-121R a a This report is a revised version of CMU-ISRI-06-121
More informationHacking-proofness and Stability in a Model of Information Security Networks
Hacking-proofness and Stability in a Model of Information Security Networks Sunghoon Hong Preliminary draft, not for citation. March 1, 2008 Abstract We introduce a model of information security networks.
More informationInfinitely Repeated Games with Discounting Ù
Infinitely Repeated Games with Discounting Page 1 Infinitely Repeated Games with Discounting Ù Introduction 1 Discounting the future 2 Interpreting the discount factor 3 The average discounted payoff 4
More information4 Learning, Regret minimization, and Equilibria
4 Learning, Regret minimization, and Equilibria A. Blum and Y. Mansour Abstract Many situations involve repeatedly making decisions in an uncertain environment: for instance, deciding what route to drive
More informationCHAPTER 9. Integer Programming
CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral
More information5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationLinear Programming in Matrix Form
Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,
More informationHow I won the Chess Ratings: Elo vs the rest of the world Competition
How I won the Chess Ratings: Elo vs the rest of the world Competition Yannis Sismanis November 2010 Abstract This article discusses in detail the rating system that won the kaggle competition Chess Ratings:
More information