Game theory for. Leonardo Badia.
|
|
- Domenic McBride
- 5 years ago
- Views:
Transcription
1 Game theory for information engineering Leonardo Badia
2 Zero-sum games A special class of games, easier to solve
3 Zero-sum We speak of zero-sum game if u i (s) = -u -i (s). player A player B L C R T -9,9 8,-8-5,5 M -2,2 6,-6 2,-2 D -1,1 3,-3 4,-4 The odd/even game and rock/paper/scissors are zero-sum games.
4 Minimax Theorem (1) Consider a zero-sum game G. (1) G has a NE iff maxmin i = minmax i for each i (2) All NEs yield the same payoff ( = maxmin i ) (3) All NEs have the form (s i*,s -i*), where s i* is a security strategy player B player A L C R T -9, 9 8, -8-5, 5 M -2, 2 6, -6 2, -2 D -1, 1 3, -3 4, -4 for player A: maxmin = -1 minmax = -1 (L,D) is a NE, u A = -1
5 Remarks Since the game is zero-sum, it is sufficient to check the maxmin = minmax condition for one player only. It also holds maxmin i = minmax -i minmax i = maxmin -i The common value of maxmin 1 = minmax 1 is called the value of the game. A joint security strategy (if any), i.e., a NE, is called a saddle point of the game.
6 Remarks The bi-matrix for this special kind of games can be represented with a regular matrix (utility of player -i is implicit). The proof of the theorem is due to von Neumann (1928) and makes use of linear programming (constrained optimization). The criterion of minmaximizing the utility has been widely employed in artificial intelligence applications: e.g., chess, which is a zero-sum (although sequential) game.
7 Case closed? In general, finding Nash Equilibria is tricky; but in the special case of zero-sum games we have a nice criterion (necessary+sufficient). However, it does not seem to be useful if the maxmin=minmax condition fails. In the odd/even game, maxmin = -4 < minmax = 4 So? Odd Even , 4 4, , -4-4, 4
8 Mixed strategies Simple games get fuzzy
9 Missing outcome Expand the Odd/Even game to find its outcome Odd Even 0 0 ½ , -4, 4 4 0, 0 4, -4 4, -4 ½ 1 0, 0 4, -4 0, 0 0, 0-4, 4 1 4, -4 0, 0-4, 4 It seems that (½,½) is a NE. Let formalize this.
10 Mixed strategies If A is a non-empty discrete set, a probability distribution over A is a function p : A [0,1], such that x A p(x) = 1 The set of possible p.d. s over A is denoted as ΔA For a normal form game (S 1,,S n ; u 1,,u n ), a mixed strategy for player i is a probability distribution m i over set S i. That is, i chooses strategies in S i = (s i,1,,s i,n ) with probabilities (m i (s i,1 ),, m i (s i,n ) ).
11 Payoff The utility function u i can be extended to a real function over ΔS 1 ΔS 2 ΔS n If players choose mixed strategies (m 1,,m n ), compute player i s payoff by weighing on m i s. u i (m 1,,m n ) = s S m 1 (s 1 ) m 2 (s 2 ) m n (s n ) u i (s) In other words: fix a global strategy s compute its probability weigh the utility of s on this probability and sum
12 Intuition Consider the odd/even game and assume Odd decides to play 0 with probability q, while Even plays 0 with probability r. Consequently 1 is played by Odd and Even with probability 1-q and 1-r, respectively. Odd 0 (prob r ) Even the table shows a single global strategies s = (q, r ) 1 (prob 1-r ) 0 (prob q) -4qr, 4qr 4q(1-r), -4q(1-r) 1 (prob 1-q) 4(1-q)r, -4(1-q)r -4(1-q)(1-r), 4(1-q)(1-r)
13 Intuition In other words, we revise the game so that each player can choose not only either 0 or 1, but also a value between them: q for Odd, r for Even. Odd s payoff is -16qr +8q +8r -4 = -4(2q-1)(2r-1) Odd 0 (prob r ) Even 1 (prob 1-r ) 0 (prob q) -4qr, 4qr 4q(1-r), -4q(1-r) 1 (prob 1-q) 4(1-q)r, -4(1-q)r -4(1-q)(1-r), 4(1-q)(1-r)
14 Intuition In other words, we revise the game so that each player can choose not only either 0 or 1, but also a value between them: q for Odd, r for Even. Odd s payoff is -16qr +8q +8r -4 = -4(2q-1)(2r-1) 0 Even 0 r 1 Odd q -16qr +8q +8r -4, 16qr -8q -8r +4 1
15 Pure strategies Given a mixed strategy m i ΔS i we define the support of m i as {s i S i : m i (s i ) > 0 } Each strategy s i S i (an element of S i ) can be identified with the mixed strategy p (which is an element of ΔS i ) such that p(s i ) = 1 Hence, p(s i ) = 0 if s i s i and also support(p)={s i } Thereafter, we identify p with s i. In this context, s i is called a pure strategy. Previous definitions of dominance and NE only refer to the pure strategy case.
16 Strict/weak dominance Consider game G = {S 1,,S n ; u 1,,u n }. If m i, m i S i, m i strictly dominates m i if u i (m i,m -i ) > u i (m i,m -i ) for every m -i We say that m i weakly dominates m i if u i (m i,m -i ) u i (m i,m -i ) for every m -i u i (m i,m -i ) > u i (m i,m -i ) for some m -i Note: there are infinitely (and continuously) many m -i in the set: S 1 S i-1 S i+1 S n
17 Strict/weak dominance However, it is possible to prove that: If m i, m i S i, m i strictly dominates m i if u i (m i,s -i ) > u i (m i,s -i ) for every s -i S -i Similarly, m i weakly dominates m i if u i (m i,s -i ) u i (m i,s -i ) for every s -i S -i u i (m i,s -i ) > u i (m i,s -i ) for some s -i S -i That is, we can limit our search to pure strategies of the opponents.
18 Nash equilibrium Consider game G = {S 1,,S n ; u 1,,u n }. A joint mixed strategy m S 1 S n is said to be a Nash equilibrium if for all i : u i (m) > u i (m i i,m -i ) for every m i i S i This reprise the same concept of NE in pure strategies: no player has an incentive to change his/her move (which is a mixed strategy now)
19 back to Example 3 In the Odd/Even game, the payoff for Odd is -4(2q - 1) (2r - 1), the opposite for Even. If q = ½, or r = ½, both players have payoff 0. If q = r = ½ no player has incentive to change. Odd Even 0 ½ 1 0 0, 0 0, 0 ½ 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 1 0, 0 Nash equilibrium
20 back to Example 3 As an exercise, prove that (½, ½) is the only Nash Equilibrium of the Odd/Even game. How to proceed. Consider three cases, where the payoff of player Odd is <0, >0, =0 but joint strategy is not (½, ½). Show that in each case there is a player (who?) having an incentive in changing strategy. None of this is a NE. (½, ½) is the only one.
21 Using mixed strategies and introducing the Nash theorem
22 IES vs mixed strategies pl layer A player B L C R T 7, 4 5, 0 8, 1 D 6, 0 3, 4 9, 1 Nash equilibrium R is not dominated by L or C. But mixed strategy m = ½L + ½C gets payoff 2 regardless of A s move. Pure strategy R is strictly dominated by m and can be eliminated. Further eliminations are possible.
23 IES vs mixed strategies Similar theorems to the pure strategy case hold for IES in mixed strategies (IESM). Theorem. Nash equilibria survive IESM. Theorem. The order of IESM is irrelevant. Note. Strict (not weak) dominance must be used. A weakly dominated strategy can be a NE, or belong to the support of a NE.
24 Characterization Theorem. Take a game G = {S 1,,S n ; u 1,,u n } and a joint mixed strategy m for game G. The following statements are equivalent: (1) Joint mixed strategy m is a Nash equilibrium. (2) For each i: u i (m) = u i (s i, m -i ) for every s i support(m i ) u i (m) u i (s i, m -i ) for every s i support(m i ) Corollary. If a pure strategy is a NE, it is such also as a mixed strategy.
25 back to Example 5 R Brian S Ann R 2, 1 0, 0 S 0, 0 1, 2 This game had two pure NEs: (R,R) and (S,S). We show now that there is also a mixed NE. Ann (or Brian) plays R with probabilities q (or r). A mixed strategy is uniquely identified by (q,r). Ann s payoff is u A (q,r) = 2qr + (1-q)(1-r) Brian s is u B (q,r) = qr + 2(1-q)(1-r)
26 back to Example 5 Assume (a,b) is a mixed NE. Note: support (a) = support (b) = {R,S}. Pure strategies R/S correspond with q (or r) being 0/1. Due to the Theorem, u A (a,b) = u A (0,b) = u A (1,b) Now, use: u A (q,r) = 2qr + (1-q)(1-r) 2ab + (1-a)(1-b) = 1-b = 2b Solution: b = ⅓. Similarly, u A (a,0) = u A (a,1) Solution: a = ⅔.
27 Nash theorem (intro) The reasoning we used to find the third (mixed) NE of the Battle of Sexes is more general. Every two-player games with two strategies has a NE in mixed strategies. This is easy to prove and is part of the more general Nash theorem. Theorem (Nash 1950). Every game with finite S i s has at least one Nash equilibrium (possibly involving mixed strategies).
28 Critics of mixed strategy Mixed strategies are important for Nash Theorem. However, many authors criticize mixed strategies as probabilities. In the end, players take a pure strategy. Possible alternative interpretations. Large numbers. Probability q means: if we play the game M times, a pure strategy is chosen qm times (note: each of the M times is one-shot memoryless) Fuzzy values. Actions are not simply white/black. Beliefs. The probability q reflects the uncertainty that my opponent has about my choice (which is pure).
29 NE as best responses Using beliefs, we can speak of best response to an opponent s (mixed) strategy. Intuition. F Bea G Art U 6, 1 0, 4 D 2, 5 4, 0 Bea ignores what Art will play, so she assumes he will play U with probability q. Similarly, Art thinks Bea will play F with probability r.
30 NE as best responses F Bea G Art U 6, 1 0, 4 D 2, 5 4, 0 E.g., if Bea is known for always playing F (r =1), Art s best response is to play U (q =1). In general? It holds: u A (D,r ) = 2r + 4(1-r ), u A (U,r ) = 6r U is actually Art s best response as long as r > ½, else it is D. If r = ½ they are equivalent. Denote Art s best response with q*(r).
31 NE as best responses q 1 q*(r ) 0 ½ 1 r For Bea: u B (q,f ) = q + 5(1-q ), u B (q,g ) = 4q Thus, Bea s best response r*(q) is step-wise r*(q) = 1 if q <⅝, r*(q) = 0 if q >⅝
32 NE as best responses q 1 ⅝ (mixed) Nash equilibrium q*(r ) r*(q ) 0 ½ 1 r Joint For Bea: strategy u B (q,f m ) = (q q =½, + 5(1-q r =⅝) ), uis B (q,g a NE. ) = 4q NE Thus, are Bea s points best were response the choice r*(q) of is each step-wise player is the best r*(q) response = 1 if q <⅝, to the other r*(q) player s = 0 if q choice. >⅝
33 Existence of NE Clearly, the existence of at least one NE is guaranteed by topological reasons. There may be more NEs (e.g. Battle of Sexes). R Brian S Ann R 2, 1 0, 0 S 0, 0 1, 2 u A (R,r ) = 2r, u A (S,r ) = 1 r, q*(r ) = 1 - h(r -⅓) u B (q,r) = q, u B (q,s) = 2(1- q), r*(q ) = 1 - h(q -⅔)
34 Existence of NE q 1 (pure) Nash equilibria ⅔ q*(r ) (mixed) Nash equilibrium r*(q ) 0 ⅓ 1 Anyway, q*(r) must intersect r*(q) at least once. The Nash theorem generalizes this reasoning. r
35 The Nash theorem Consider game G = {S 1,,S n ; u 1,,u n }. Define: best i : ΔS 1 ΔS i-1 ΔS i+1 ΔS n ΔS i best i (m -i ) = {m i ΔS i : u i (m i, m -i ) is maximal } Then define best : ΔS ΔS as best(m) = best 1 (m -1 ) best n (m -n ) That is, best i (m -i ) is the set of best responses of user i to moves m -i by other players. Aggregating them, we obtain best. m is a NE if m best(m) Note. best i (m -i ) is always non-empty and always contains at least a pure strategy.
36 The Nash theorem Lemma (Kakutani Theorem). Let A be a compact and convex subset of R n. If F : A A is s.t.: For all x A, F(x) is non-empty and convex Let {x i } and {y i } be sequences converging to x and y, respectively. If y i F(x i ) then y F(x). Then there exists x* A such that x* F(x*). Nash theorem. It s nothing but Kakutani theorem applied to previously defined function best.
37 Mixed maxmin/minmax the extensions to mixed strategies
38 Mixed security strategy Consider a two- player game ( i vs -i ), and take f i :ΔS i R as f i (m i ) = min m-i ΔS -i u i (m i,m -i ) Any mixed strategy m i * maximizing f i (m i ) is a mixed security strategy for i. This max, i.e. max mi ΔS i min m-i Δ S -i u i (m i,m -i ) is the maxmin im or the mixed security payoff of i. A mixed security strategy is the conservative mixed strategy guaranteeing the highest payoff for i in case of the worst mixed strategy by -i.
39 Mixed minmax Also if F i :ΔS -i R is F i (m -i ) = max mi ΔS i u i (m i,m -i ) min m-i ΔS -i F i (m -i ) = min m-i ΔS -i max mi ΔS i u i (m i,m -i ) is the minmax for i in mixed strategy, minmax im. If i could move after -i, there is a mixed strategy which guarantees i to achieve at least minmax m i. Note 1. It can be shown that f i (m i ) can be found minimizing u i (m i,s -i ), i.e., using pure strategies only. Equally, F i (m -i ) can be defined maximizing u i (s i,m -i ) Note 2. maxmin im and minmax im always exist. This is due to payoff u i (m i,m -i ) being continuous.
40 maxmin m vs minmax m We already know from pure minmax: (1) For every player i, maxmin im minmax i m (2) If joint mixed strategy m is a Nash equilibrium, then for every player i, minmax im u i (m) Jim S Joe C T 3,- 0,- M 1,- 2,- (only Jim s payoffs are shown) Jim: maxmin = 1, minmax = 2 Jim can increase his maxmin if he plays ¼ T + ¾ M. maxmin m = 1.5 For Jim, the worst strategy Joe can play is ⅓ S + ⅔ C, minmax m = 1.5 i i
41 maxmin m vs minmax m Art Bea F G U 6, 1 0, 4 D 2, 5 4, maxmin m ¼ ½ 1 q Art s mixed strategies are uniquely described by q. f A (q) = min sb {F,G} u A (q,s B ) = min { u A (q,f), u A (q,g) } = = min { 6q+2(1-q), 4(1-q) } = min { 2+4q, 4-4q }
42 maxmin m vs minmax m Check yourself that minmax Am is also 3. So it is verified that maxmin i maxmin im minmax im minmax i As an exercise, do the same check for Bea. Note that we found a Nash equilibrium at (⅝, ½), so Art s payoff at NE is 3.75 > 3.
43 back to Example 3 Odd 0 Even 1 0-4, 4 4, , -4-4, 4 Also for this game (which is zero-sum) maxmin = -4 < maxmin im = minmax im = 0 < minmax = 4 Condition maxmin im = minmax im seems to hold. 0 was the payoff at the (mixed) NE for this zerosum game. Remember the minimax theorem?
44 Minimax Theorem (2) (1) For every player i, maxmin im = minmax i m (2) If G is a zero-sum game, all Nash equilibria in mixed strategies are security strategies for player i and yield a payoff to player i equal to maxmin m i Note. In zero-sum games maxmin 1 m = -minmax 2 m All Nash equilibria are equivalent (same payoff) maxmin 1 m is called the value of the game.
45 Linear Programming The search of minmax solutions (i.e., NEs) of a zero-sum game is a nice application of LP. Player 1 has pure strategies {A 1, A 2,, A L }. A mixed strategy a = { a j } is a linear combination a 1 A a L A L Player 2 has pure strategies {B 1, B 2,, B M }. A mixed strategy b = { b j } is a linear combination b 1 B b M B M Note. We only need u = u 1 as u 2 = - u 1
46 Linear Programming a j 0, j a j = 1 j a j u(a j B k ) W ( k) maximize W W must be maximized. W cannot be increased, when some constraints become active. These constraint describe the support of player 2 s b. The a j s are a probability distribution M constraints The payoff of (a, B k ). We check a against M pure strategies only In general, this finds a mixed minimax strategy for player 1.
47 Linear Programming Since maxmin im = minmax i m a j 0, j a j = 1 j a j u(a j B k ) W ( k) maximize W minmax version b j 0, j b j = 1 j b j u(a k B j ) W ( k) minimize W maxmin version The two problems yield the same solution. Note. This formulation can be made for every problem, but solution is not always guaranteed. Zero-sum games are special in that u 2 = - u 1
48 How to solve minmax LP problems can be solved by using well known techniques (see Optimization courses). Polynomial-time techniques exist. Simplex method is widely used (CPLEX, lpsolve). Even though (worst-case) exponential, it is often fast in practice. Meta-heuristic techniques (Genetic Algorithms, Tabu search): sometimes even faster, but they do not guarantee to find the solution.
6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts
6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 9, 2010 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria
More informationGame theory and applications: Lecture 1
Game theory and applications: Lecture 1 Adam Szeidl September 20, 2018 Outline for today 1 Some applications of game theory 2 Games in strategic form 3 Dominance 4 Nash equilibrium 1 / 8 1. Some applications
More informationRegret Minimization and Security Strategies
Chapter 5 Regret Minimization and Security Strategies Until now we implicitly adopted a view that a Nash equilibrium is a desirable outcome of a strategic game. In this chapter we consider two alternative
More information10.1 Elimination of strictly dominated strategies
Chapter 10 Elimination by Mixed Strategies The notions of dominance apply in particular to mixed extensions of finite strategic games. But we can also consider dominance of a pure strategy by a mixed strategy.
More information6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2
6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria Mixed Strategies
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 22 COOPERATIVE GAME THEORY Correlated Strategies and Correlated
More informationBasic Game-Theoretic Concepts. Game in strategic form has following elements. Player set N. (Pure) strategy set for player i, S i.
Basic Game-Theoretic Concepts Game in strategic form has following elements Player set N (Pure) strategy set for player i, S i. Payoff function f i for player i f i : S R, where S is product of S i s.
More informationAdvanced Microeconomics
Advanced Microeconomics ECON5200 - Fall 2014 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market
More informationIntroduction to game theory LECTURE 2
Introduction to game theory LECTURE 2 Jörgen Weibull February 4, 2010 Two topics today: 1. Existence of Nash equilibria (Lecture notes Chapter 10 and Appendix A) 2. Relations between equilibrium and rationality
More informationBest-Reply Sets. Jonathan Weinstein Washington University in St. Louis. This version: May 2015
Best-Reply Sets Jonathan Weinstein Washington University in St. Louis This version: May 2015 Introduction The best-reply correspondence of a game the mapping from beliefs over one s opponents actions to
More informationRepeated Games. September 3, Definitions: Discounting, Individual Rationality. Finitely Repeated Games. Infinitely Repeated Games
Repeated Games Frédéric KOESSLER September 3, 2007 1/ Definitions: Discounting, Individual Rationality Finitely Repeated Games Infinitely Repeated Games Automaton Representation of Strategies The One-Shot
More informationMATH 121 GAME THEORY REVIEW
MATH 121 GAME THEORY REVIEW ERIN PEARSE Contents 1. Definitions 2 1.1. Non-cooperative Games 2 1.2. Cooperative 2-person Games 4 1.3. Cooperative n-person Games (in coalitional form) 6 2. Theorems and
More informationCS711: Introduction to Game Theory and Mechanism Design
CS711: Introduction to Game Theory and Mechanism Design Teacher: Swaprava Nath Domination, Elimination of Dominated Strategies, Nash Equilibrium Domination Normal form game N, (S i ) i N, (u i ) i N Definition
More informationOutline for today. Stat155 Game Theory Lecture 13: General-Sum Games. General-sum games. General-sum games. Dominated pure strategies
Outline for today Stat155 Game Theory Lecture 13: General-Sum Games Peter Bartlett October 11, 2016 Two-player general-sum games Definitions: payoff matrices, dominant strategies, safety strategies, Nash
More informationStrategies and Nash Equilibrium. A Whirlwind Tour of Game Theory
Strategies and Nash Equilibrium A Whirlwind Tour of Game Theory (Mostly from Fudenberg & Tirole) Players choose actions, receive rewards based on their own actions and those of the other players. Example,
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India August 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India August 2012 Chapter 6: Mixed Strategies and Mixed Strategy Nash Equilibrium
More informationUsing the Maximin Principle
Using the Maximin Principle Under the maximin principle, it is easy to see that Rose should choose a, making her worst-case payoff 0. Colin s similar rationality as a player induces him to play (under
More informationIntroduction to Multi-Agent Programming
Introduction to Multi-Agent Programming 10. Game Theory Strategic Reasoning and Acting Alexander Kleiner and Bernhard Nebel Strategic Game A strategic game G consists of a finite set N (the set of players)
More informationComparative Study between Linear and Graphical Methods in Solving Optimization Problems
Comparative Study between Linear and Graphical Methods in Solving Optimization Problems Mona M Abd El-Kareem Abstract The main target of this paper is to establish a comparative study between the performance
More informationGame Theory: Normal Form Games
Game Theory: Normal Form Games Michael Levet June 23, 2016 1 Introduction Game Theory is a mathematical field that studies how rational agents make decisions in both competitive and cooperative situations.
More informationCS 331: Artificial Intelligence Game Theory I. Prisoner s Dilemma
CS 331: Artificial Intelligence Game Theory I 1 Prisoner s Dilemma You and your partner have both been caught red handed near the scene of a burglary. Both of you have been brought to the police station,
More informationSolution to Tutorial 1
Solution to Tutorial 1 011/01 Semester I MA464 Game Theory Tutor: Xiang Sun August 4, 011 1 Review Static means one-shot, or simultaneous-move; Complete information means that the payoff functions are
More informationSolution to Tutorial /2013 Semester I MA4264 Game Theory
Solution to Tutorial 1 01/013 Semester I MA464 Game Theory Tutor: Xiang Sun August 30, 01 1 Review Static means one-shot, or simultaneous-move; Complete information means that the payoff functions are
More informationWarm Up Finitely Repeated Games Infinitely Repeated Games Bayesian Games. Repeated Games
Repeated Games Warm up: bargaining Suppose you and your Qatz.com partner have a falling-out. You agree set up two meetings to negotiate a way to split the value of your assets, which amount to $1 million
More information1 Games in Strategic Form
1 Games in Strategic Form A game in strategic form or normal form is a triple Γ (N,{S i } i N,{u i } i N ) in which N = {1,2,...,n} is a finite set of players, S i is the set of strategies of player i,
More informationEpistemic Game Theory
Epistemic Game Theory Lecture 1 ESSLLI 12, Opole Eric Pacuit Olivier Roy TiLPS, Tilburg University MCMP, LMU Munich ai.stanford.edu/~epacuit http://olivier.amonbofis.net August 6, 2012 Eric Pacuit and
More informationChapter 10: Mixed strategies Nash equilibria, reaction curves and the equality of payoffs theorem
Chapter 10: Mixed strategies Nash equilibria reaction curves and the equality of payoffs theorem Nash equilibrium: The concept of Nash equilibrium can be extended in a natural manner to the mixed strategies
More informationGAME THEORY. Game theory. The odds and evens game. Two person, zero sum game. Prototype example
Game theory GAME THEORY (Hillier & Lieberman Introduction to Operations Research, 8 th edition) Mathematical theory that deals, in an formal, abstract way, with the general features of competitive situations
More informationMath 167: Mathematical Game Theory Instructor: Alpár R. Mészáros
Math 167: Mathematical Game Theory Instructor: Alpár R. Mészáros Midterm #1, February 3, 2017 Name (use a pen): Student ID (use a pen): Signature (use a pen): Rules: Duration of the exam: 50 minutes. By
More informationChair of Communications Theory, Prof. Dr.-Ing. E. Jorswieck. Übung 5: Supermodular Games
Chair of Communications Theory, Prof. Dr.-Ing. E. Jorswieck Übung 5: Supermodular Games Introduction Supermodular games are a class of non-cooperative games characterized by strategic complemetariteis
More informationCan we have no Nash Equilibria? Can you have more than one Nash Equilibrium? CS 430: Artificial Intelligence Game Theory II (Nash Equilibria)
CS 0: Artificial Intelligence Game Theory II (Nash Equilibria) ACME, a video game hardware manufacturer, has to decide whether its next game machine will use DVDs or CDs Best, a video game software producer,
More informationGame Theory: introduction and applications to computer networks
Game Theory: introduction and applications to computer networks Zero-Sum Games (follow-up) Giovanni Neglia INRIA EPI Maestro 20 January 2014 Part of the slides are based on a previous course with D. Figueiredo
More informationChapter 2 Strategic Dominance
Chapter 2 Strategic Dominance 2.1 Prisoner s Dilemma Let us start with perhaps the most famous example in Game Theory, the Prisoner s Dilemma. 1 This is a two-player normal-form (simultaneous move) game.
More informationm 11 m 12 Non-Zero Sum Games Matrix Form of Zero-Sum Games R&N Section 17.6
Non-Zero Sum Games R&N Section 17.6 Matrix Form of Zero-Sum Games m 11 m 12 m 21 m 22 m ij = Player A s payoff if Player A follows pure strategy i and Player B follows pure strategy j 1 Results so far
More informationEconomics 171: Final Exam
Question 1: Basic Concepts (20 points) Economics 171: Final Exam 1. Is it true that every strategy is either strictly dominated or is a dominant strategy? Explain. (5) No, some strategies are neither dominated
More informationYao s Minimax Principle
Complexity of algorithms The complexity of an algorithm is usually measured with respect to the size of the input, where size may for example refer to the length of a binary word describing the input,
More informationCS 7180: Behavioral Modeling and Decision- making in AI
CS 7180: Behavioral Modeling and Decision- making in AI Algorithmic Game Theory Prof. Amy Sliva November 30, 2012 Prisoner s dilemma Two criminals are arrested, and each offered the same deal: If you defect
More informationCS711 Game Theory and Mechanism Design
CS711 Game Theory and Mechanism Design Problem Set 1 August 13, 2018 Que 1. [Easy] William and Henry are participants in a televised game show, seated in separate booths with no possibility of communicating
More informationIntroductory Microeconomics
Prof. Wolfram Elsner Faculty of Business Studies and Economics iino Institute of Institutional and Innovation Economics Introductory Microeconomics More Formal Concepts of Game Theory and Evolutionary
More informationMicroeconomics II. CIDE, MsC Economics. List of Problems
Microeconomics II CIDE, MsC Economics List of Problems 1. There are three people, Amy (A), Bart (B) and Chris (C): A and B have hats. These three people are arranged in a room so that B can see everything
More informationCHAPTER 14: REPEATED PRISONER S DILEMMA
CHAPTER 4: REPEATED PRISONER S DILEMMA In this chapter, we consider infinitely repeated play of the Prisoner s Dilemma game. We denote the possible actions for P i by C i for cooperating with the other
More informationAn introduction on game theory for wireless networking [1]
An introduction on game theory for wireless networking [1] Ning Zhang 14 May, 2012 [1] Game Theory in Wireless Networks: A Tutorial 1 Roadmap 1 Introduction 2 Static games 3 Extensive-form games 4 Summary
More informationGame Theory. Analyzing Games: From Optimality to Equilibrium. Manar Mohaisen Department of EEC Engineering
Game Theory Analyzing Games: From Optimality to Equilibrium Manar Mohaisen Department of EEC Engineering Korea University of Technology and Education (KUT) Content Optimality Best Response Domination Nash
More informationMA200.2 Game Theory II, LSE
MA200.2 Game Theory II, LSE Answers to Problem Set [] In part (i), proceed as follows. Suppose that we are doing 2 s best response to. Let p be probability that player plays U. Now if player 2 chooses
More informationMAT 4250: Lecture 1 Eric Chung
1 MAT 4250: Lecture 1 Eric Chung 2Chapter 1: Impartial Combinatorial Games 3 Combinatorial games Combinatorial games are two-person games with perfect information and no chance moves, and with a win-or-lose
More informationd. Find a competitive equilibrium for this economy. Is the allocation Pareto efficient? Are there any other competitive equilibrium allocations?
Answers to Microeconomics Prelim of August 7, 0. Consider an individual faced with two job choices: she can either accept a position with a fixed annual salary of x > 0 which requires L x units of labor
More information(a) Describe the game in plain english and find its equivalent strategic form.
Risk and Decision Making (Part II - Game Theory) Mock Exam MIT/Portugal pages Professor João Soares 2007/08 1 Consider the game defined by the Kuhn tree of Figure 1 (a) Describe the game in plain english
More informationBRIEF INTRODUCTION TO GAME THEORY
BRIEF INTRODUCTION TO GAME THEORY Game Theory Mathematical theory that deals with the general features of competitive situations. Examples: parlor games, military battles, political campaigns, advertising
More informationComplexity of Iterated Dominance and a New Definition of Eliminability
Complexity of Iterated Dominance and a New Definition of Eliminability Vincent Conitzer and Tuomas Sandholm Carnegie Mellon University 5000 Forbes Avenue Pittsburgh, PA 15213 {conitzer, sandholm}@cs.cmu.edu
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 2 1. Consider a zero-sum game, where
More informationLecture 1: Normal Form Games: Refinements and Correlated Equilibrium
Lecture 1: Normal Form Games: Refinements and Correlated Equilibrium Albert Banal-Estanol April 2006 Lecture 1 2 Albert Banal-Estanol Trembling hand perfect equilibrium: Motivation, definition and examples
More informationIn the Name of God. Sharif University of Technology. Microeconomics 2. Graduate School of Management and Economics. Dr. S.
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 2 44706 (1394-95 2 nd term) - Group 2 Dr. S. Farshad Fatemi Chapter 8: Simultaneous-Move Games
More informationPAULI MURTO, ANDREY ZHUKOV
GAME THEORY SOLUTION SET 1 WINTER 018 PAULI MURTO, ANDREY ZHUKOV Introduction For suggested solution to problem 4, last year s suggested solutions by Tsz-Ning Wong were used who I think used suggested
More informationGAME THEORY. Department of Economics, MIT, Follow Muhamet s slides. We need the following result for future reference.
14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Existence and Continuity of Nash Equilibria Follow Muhamet s slides. We need the following result for future reference. Theorem 1. Suppose
More informationECE 586GT: Problem Set 1: Problems and Solutions Analysis of static games
University of Illinois Fall 2018 ECE 586GT: Problem Set 1: Problems and Solutions Analysis of static games Due: Tuesday, Sept. 11, at beginning of class Reading: Course notes, Sections 1.1-1.4 1. [A random
More informationGAME THEORY. (Hillier & Lieberman Introduction to Operations Research, 8 th edition)
GAME THEORY (Hillier & Lieberman Introduction to Operations Research, 8 th edition) Game theory Mathematical theory that deals, in an formal, abstract way, with the general features of competitive situations
More informationMixed Strategies. In the previous chapters we restricted players to using pure strategies and we
6 Mixed Strategies In the previous chapters we restricted players to using pure strategies and we postponed discussing the option that a player may choose to randomize between several of his pure strategies.
More informationNotes on Game Theory Debasis Mishra October 29, 2018
Notes on Game Theory Debasis Mishra October 29, 2018 1 1 Games in Strategic Form A game in strategic form or normal form is a triple Γ (N,{S i } i N,{u i } i N ) in which N = {1,2,...,n} is a finite set
More informationCOS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture #24 Scribe: Jordan Ash May 1, 2014
COS 5: heoretical Machine Learning Lecturer: Rob Schapire Lecture #24 Scribe: Jordan Ash May, 204 Review of Game heory: Let M be a matrix with all elements in [0, ]. Mindy (called the row player) chooses
More informationIterated Dominance and Nash Equilibrium
Chapter 11 Iterated Dominance and Nash Equilibrium In the previous chapter we examined simultaneous move games in which each player had a dominant strategy; the Prisoner s Dilemma game was one example.
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 3 1. Consider the following strategic
More informationCS 798: Homework Assignment 4 (Game Theory)
0 5 CS 798: Homework Assignment 4 (Game Theory) 1.0 Preferences Assigned: October 28, 2009 Suppose that you equally like a banana and a lottery that gives you an apple 30% of the time and a carrot 70%
More informationIntroduction to Game Theory Lecture Note 5: Repeated Games
Introduction to Game Theory Lecture Note 5: Repeated Games Haifeng Huang University of California, Merced Repeated games Repeated games: given a simultaneous-move game G, a repeated game of G is an extensive
More informationGeneral Equilibrium under Uncertainty
General Equilibrium under Uncertainty The Arrow-Debreu Model General Idea: this model is formally identical to the GE model commodities are interpreted as contingent commodities (commodities are contingent
More informationProblem Set 2 - SOLUTIONS
Problem Set - SOLUTONS 1. Consider the following two-player game: L R T 4, 4 1, 1 B, 3, 3 (a) What is the maxmin strategy profile? What is the value of this game? Note, the question could be solved like
More informationIntroduction to Game Theory
Introduction to Game Theory 3a. More on Normal-Form Games Dana Nau University of Maryland Nau: Game Theory 1 More Solution Concepts Last time, we talked about several solution concepts Pareto optimality
More informationMATH 4321 Game Theory Solution to Homework Two
MATH 321 Game Theory Solution to Homework Two Course Instructor: Prof. Y.K. Kwok 1. (a) Suppose that an iterated dominance equilibrium s is not a Nash equilibrium, then there exists s i of some player
More informationEcon 101A Final exam May 14, 2013.
Econ 101A Final exam May 14, 2013. Do not turn the page until instructed to. Do not forget to write Problems 1 in the first Blue Book and Problems 2, 3 and 4 in the second Blue Book. 1 Econ 101A Final
More informationSF2972 GAME THEORY Infinite games
SF2972 GAME THEORY Infinite games Jörgen Weibull February 2017 1 Introduction Sofar,thecoursehasbeenfocusedonfinite games: Normal-form games with a finite number of players, where each player has a finite
More informationIn the Name of God. Sharif University of Technology. Graduate School of Management and Economics
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics (for MBA students) 44111 (1393-94 1 st term) - Group 2 Dr. S. Farshad Fatemi Game Theory Game:
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Bargaining We will now apply the concept of SPNE to bargaining A bit of background Bargaining is hugely interesting but complicated to model It turns out that the
More informationFinding Mixed-strategy Nash Equilibria in 2 2 Games ÙÛ
Finding Mixed Strategy Nash Equilibria in 2 2 Games Page 1 Finding Mixed-strategy Nash Equilibria in 2 2 Games ÙÛ Introduction 1 The canonical game 1 Best-response correspondences 2 A s payoff as a function
More informationAlgorithmic Game Theory and Applications. Lecture 11: Games of Perfect Information
Algorithmic Game Theory and Applications Lecture 11: Games of Perfect Information Kousha Etessami finite games of perfect information Recall, a perfect information (PI) game has only 1 node per information
More informationGame Theory Tutorial 3 Answers
Game Theory Tutorial 3 Answers Exercise 1 (Duality Theory) Find the dual problem of the following L.P. problem: max x 0 = 3x 1 + 2x 2 s.t. 5x 1 + 2x 2 10 4x 1 + 6x 2 24 x 1 + x 2 1 (1) x 1 + 3x 2 = 9 x
More informationTopics in Contract Theory Lecture 3
Leonardo Felli 9 January, 2002 Topics in Contract Theory Lecture 3 Consider now a different cause for the failure of the Coase Theorem: the presence of transaction costs. Of course for this to be an interesting
More informationOutline Introduction Game Representations Reductions Solution Concepts. Game Theory. Enrico Franchi. May 19, 2010
May 19, 2010 1 Introduction Scope of Agent preferences Utility Functions 2 Game Representations Example: Game-1 Extended Form Strategic Form Equivalences 3 Reductions Best Response Domination 4 Solution
More informationGame Theory Problem Set 4 Solutions
Game Theory Problem Set 4 Solutions 1. Assuming that in the case of a tie, the object goes to person 1, the best response correspondences for a two person first price auction are: { }, < v1 undefined,
More informationRationalizable Strategies
Rationalizable Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Jun 1st, 2015 C. Hurtado (UIUC - Economics) Game Theory On the Agenda 1
More informationECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves
University of Illinois Spring 01 ECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves Due: Reading: Thursday, April 11 at beginning of class
More informationAnswers to Problem Set 4
Answers to Problem Set 4 Economics 703 Spring 016 1. a) The monopolist facing no threat of entry will pick the first cost function. To see this, calculate profits with each one. With the first cost function,
More informationMicroeconomics of Banking: Lecture 5
Microeconomics of Banking: Lecture 5 Prof. Ronaldo CARPIO Oct. 23, 2015 Administrative Stuff Homework 2 is due next week. Due to the change in material covered, I have decided to change the grading system
More informationTR : Knowledge-Based Rational Decisions
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2009 TR-2009011: Knowledge-Based Rational Decisions Sergei Artemov Follow this and additional works
More informationUncertainty in Equilibrium
Uncertainty in Equilibrium Larry Blume May 1, 2007 1 Introduction The state-preference approach to uncertainty of Kenneth J. Arrow (1953) and Gérard Debreu (1959) lends itself rather easily to Walrasian
More information1 Solutions to Homework 4
1 Solutions to Homework 4 1.1 Q1 Let A be the event that the contestant chooses the door holding the car, and B be the event that the host opens a door holding a goat. A is the event that the contestant
More informationFinding Equilibria in Games of No Chance
Finding Equilibria in Games of No Chance Kristoffer Arnsfelt Hansen, Peter Bro Miltersen, and Troels Bjerre Sørensen Department of Computer Science, University of Aarhus, Denmark {arnsfelt,bromille,trold}@daimi.au.dk
More informationRepeated Games. EC202 Lectures IX & X. Francesco Nava. January London School of Economics. Nava (LSE) EC202 Lectures IX & X Jan / 16
Repeated Games EC202 Lectures IX & X Francesco Nava London School of Economics January 2011 Nava (LSE) EC202 Lectures IX & X Jan 2011 1 / 16 Summary Repeated Games: Definitions: Feasible Payoffs Minmax
More informationMS&E 246: Lecture 5 Efficiency and fairness. Ramesh Johari
MS&E 246: Lecture 5 Efficiency and fairness Ramesh Johari A digression In this lecture: We will use some of the insights of static game analysis to understand efficiency and fairness. Basic setup N players
More informationGame Theory for Wireless Engineers Chapter 3, 4
Game Theory for Wireless Engineers Chapter 3, 4 Zhongliang Liang ECE@Mcmaster Univ October 8, 2009 Outline Chapter 3 - Strategic Form Games - 3.1 Definition of A Strategic Form Game - 3.2 Dominated Strategies
More informationElements of Economic Analysis II Lecture X: Introduction to Game Theory
Elements of Economic Analysis II Lecture X: Introduction to Game Theory Kai Hao Yang 11/14/2017 1 Introduction and Basic Definition of Game So far we have been studying environments where the economic
More information6.207/14.15: Networks Lecture 9: Introduction to Game Theory 1
6.207/14.15: Networks Lecture 9: Introduction to Game Theory 1 Daron Acemoglu and Asu Ozdaglar MIT October 13, 2009 1 Introduction Outline Decisions, Utility Maximization Games and Strategies Best Responses
More informationIntroduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4)
Introduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4) Outline: Modeling by means of games Normal form games Dominant strategies; dominated strategies,
More informationGame Theory. Wolfgang Frimmel. Repeated Games
Game Theory Wolfgang Frimmel Repeated Games 1 / 41 Recap: SPNE The solution concept for dynamic games with complete information is the subgame perfect Nash Equilibrium (SPNE) Selten (1965): A strategy
More informationFebruary 23, An Application in Industrial Organization
An Application in Industrial Organization February 23, 2015 One form of collusive behavior among firms is to restrict output in order to keep the price of the product high. This is a goal of the OPEC oil
More informationEcon 618: Topic 11 Introduction to Coalitional Games
Econ 618: Topic 11 Introduction to Coalitional Games Sunanda Roy 1 Coalitional games with transferable payoffs, the Core Consider a game with a finite set of players. A coalition is a nonempty subset of
More informationLecture 5: Iterative Combinatorial Auctions
COMS 6998-3: Algorithmic Game Theory October 6, 2008 Lecture 5: Iterative Combinatorial Auctions Lecturer: Sébastien Lahaie Scribe: Sébastien Lahaie In this lecture we examine a procedure that generalizes
More informationHW Consider the following game:
HW 1 1. Consider the following game: 2. HW 2 Suppose a parent and child play the following game, first analyzed by Becker (1974). First child takes the action, A 0, that produces income for the child,
More informationEcon 618 Simultaneous Move Bayesian Games
Econ 618 Simultaneous Move Bayesian Games Sunanda Roy 1 The Bayesian game environment A game of incomplete information or a Bayesian game is a game in which players do not have full information about each
More informationJianfei Shen. School of Economics, The University of New South Wales, Sydney 2052, Australia
. Zero-sum games Jianfei Shen School of Economics, he University of New South Wales, Sydney, Australia emember that in a zerosum game, u.s ; s / C u.s ; s / D, s ; s. Exercise. Step efer Matrix A, we know
More informationStrategy Lines and Optimal Mixed Strategy for R
Strategy Lines and Optimal Mixed Strategy for R Best counterstrategy for C for given mixed strategy by R In the previous lecture we saw that if R plays a particular mixed strategy, [p, p, and shows no
More informationCUR 412: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 2015
CUR 41: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 015 Instructions: Please write your name in English. This exam is closed-book. Total time: 10 minutes. There are 4 questions,
More information