A course in game theory

Page numbers in boldface indicate pages on which objects are defined.

A B C D E F G H I J K L M N O P Q R S T U V W Z

A

• Action
  • extensive game with imperfect information, Definition 200.1
  • extensive game with perfect information, Definition 89.1
  • strategic game, 11
• Action-determined beliefs, Definition 232.1
• ADD (Shapley value axiom), 292
• Addition of a superfluous move
  • equivalence of extensive games, 206
• Additive coalitional game, Exercise 261.2
• Additivity axiom of Shapley value, 292
• Agent strategic form, 250
• Agree to disagree, 75
• Agreements, set of
  • bargaining game, 118
  • Nash solution, Definition 300.1
• Air strike, Exercise 36.2
• Allocation
  • competitive, 269
  • market with transferable payoff, 264
• Alternating offers bargaining, see bargaining game of alternating offers
• Approachable mixed strategy equilibrium, 42
• Armies, Exercise 101.3
• Assessment, 221, Definition 222.1
  • consistent, Definition 224.2
• Asymmetric abilities/perceptions, 6
• Asymmetric Nash solution, 309
• Auction, Example 18.1
  • first price, Exercise 18.2
  • second price, Exercise 18.3
    • imperfect information, Example 27.1
• Automaton, see machine
• Axiomatizations of the core, 275
• Axioms
  • for knowledge function, 69-70
  • for Nash solution, 305-307
  • for Shapley value, 292

B

• Bach or Stravinsky?, Example 15.3
  • Bayesian game, Exercise 27.2
  • correlated equilibrium, 44, Example 46.1
  • interpretation of mixed equilibrium, 40
  • mixed extension, Example 34.1
  • Nash equilibrium, Example 15.3
  • with outside option, Example 110.1
• Backwards induction, 99
• Balanced contributions property for coalitional games, Definition 291.1
• Balanced game/collection of weights, 262
• Balancing counterobjection, see counterobjection
• Bargaining game of alternating offers, 120
  • constant cost of delay, Exercise 125.2
  • equilibria with delay, Exercise 125.2
  • importance of procedure, 127
  • more than two players, 130
  • multiple equilibrium outcomes, Exercise 125.2, Exercise 128.1
  • Nash equilibria, 121
  • and Nash solution, 310-311
  • opting out, 128
  • pie with discrete elements, Exercise 128.1
  • properties of subgame perfect equilibria, 125
  • risk of breakdown, 129
    • relation with Nash solution, Proposition 310.3
  • subgame perfect equilibria, 121-127
    • characterization, Proposition 122.1
  • variant for coalitional game, Exercise 296.1
  • variant with one party proposing, Exercise 127.1
  • variant with simultaneous proposals, 127
• Bargaining problem (Nash), Definition 300.1
• Bargaining set of coalitional game, Definition 282.1
  • and kernel, Lemma 285.1
• Bargaining solution, Definition 301.1, 308
  • Nash, Definition 301.2
• Battle of the sexes, see Bach or Stravinsky?
• Bayesian extensive game, Definition 231.1
  • perfect Bayesian equilibrium, Definition 232.1
• Bayesian game, Definition 25.1
  • common prior belief, 75
  • extensive with observable actions, Definition 231.1
  • Nash equilibrium, Definition 26.1
  • signaling game, 237
• Beer or Quiche, Example 244.2
• Behavioral strategy, Definition 212.1
  • completely mixed, 224
  • Nash equilibrium, 216
  • outcome-equivalence with mixed strategy, 214
• Belief system, 223
• Beliefs
  • constraints in extensive game
    • common beliefs, 222
    • consistency with strategies, 221
    • structural consistency, 222, Definition 228.1
  • mixed strategies, 43
  • rationalizability, 54
• Best response function, 15
• Best response to belief
  • action not strictly dominated, Lemma 60.1
  • action not weakly dominated, Exercise 64.2
• Biological example, Example 49.2, 49
• Bondareva-Shapley theorem, Proposition 262.1
• BoS, see Bach or Stravinsky?
• Bounded rationality, 6, 164
• Breakdown in bargaining, 129
• Burning money game, Example 111.1

C

• Card game, Exercise 217.3
• Centipede game, 106
• Chain-store game, 105
  • perturbation, Example 239.1
• Chance moves in extensive game
  • imperfect information, 201
  • perfect information, 101
• Chess, 6, 100
• Chicken, 30
• Choice rule/function, 178
  • monotonic, Definition 186.1
  • no veto power, Definition 187.1
• Choice theory, 4
• Choice under uncertainty, 4, Exercise 71.2
• Clarke-Groves game forms, 183
• Coalescing of moves
  • equivalence of extensive games, 207
  • sequential equilibrium, Example 226.2
  • trembling hand perfect equilibrium, Exercise 252.2
• Coalition, Definition 257.1
• Coalitional game
  • nontransferable payoff, Definition 268.2
    • bargaining set, kernel, 284, 298
    • core, Definition 268.3
    • exchange economy, 269
    • Shapley value, 298
  • solutions, see solution
  • transferable payoff, Definition 257.1
    • balanced, 262
    • balanced contributions property, Definition 291.1
    • bargaining set, Definition 282.1
    • cohesive, Definition 258.1
    • convex, Exercise 260.4
    • core, Definition 258.2
    • cost-sharing, 296
    • excess of a coalition, 283
    • feasible payoff vector/profile, 258
    • imputation, 278
    • kernel, Definition 284.1
    • nonemptiness of bargaining set and kernel, Corollary 288.3
    • nonemptiness of nucleolus, Proposition 288.1
    • nonempty core, Proposition 262.1
    • nucleolus, Definition 286.1
    • objection, see objection, counterobjection
    • Shapley value, Definition 291.2
    • simple, Exercise 261.1
    • stable set, Definition 279.1
    • superadditive, 258
    • zerosum, Exercise 261.2
• Coalitional vs. noncooperative games, 2, 255
• Cohesive coalitional game, Definition 258.1
• Common beliefs, 222
• Common knowledge, Definition 73.1, Definition 73.2
• Communication, 113
• Comparative statics of risk aversion in Nash bargaining, Proposition 304.1
• Competitive equilibrium
  • exchange economy, 269
    • and core, 271-274
    • existence, 270
  • and game theory, 3
  • market with transferable payoff, 266
    • and core, 265-268
    • existence, Exercise 267.2
• Competitive payoff, 266
• Completely mixed strategy, 224
• Complexity of machine, 165
• Concave function, 7
• Consequences, set of
  • coalitional game, Definition 268.2
  • implementation theory, 178
  • strategic game, 12
• Consistency
  • of beliefs with strategies, 221
  • of strategy with history, 213
  • structural, Definition 228.1
• Consistent assessment, Definition 224.2
  • and structural consistency, 228-231
• Constituent game of repeated game, 136
• Constraints on beliefs, 221-222
• Continuous preference relation, 7
• Convergence of core and competitive equilibrium
  • exchange economy, Proposition 273.1
  • market with transferable payoff, 267
• Convex coalitional game, Exercise 260.4
  • core and Shapley value, Exercise 295.5
• Convexity of Nash bargaining problem, Definition 300.1
• Cooperative game, see coalitional game
• Coordinated attack problem, 85
• Coordination game, Example 16.1
• Core
  • convergence to competitive equilibrium, 267, Proposition 273.1
  • equal treatment in, Lemma 272.2
  • exchange economy, 271
    • and competitive equilibrium, 271-274
  • and generalized stable set, Exercise 280.5
  • market with transferable payoff, 264
    • and competitive equilibrium, 265-268
  • nontransferable payoff game, Definition 268.3
  • and stable sets, Proposition 279.2
  • transferable payoff game, Definition 258.2
    • nonemptiness, Proposition 262.1
• Correlated equilibrium, Definition 45.1
  • knowledge, Exercise 81.1
  • rationalizability, Lemma 56.2
• Cost-sharing in coalitional game, 296
• Counterobjection in coalitional game
  • bargaining set, 281
  • kernel, 284
  • Nash solution, 302
  • nucleolus, 286
  • Shapley value, 290
• Cournot duopoly
  • rationalizability, Exercise 56.4
• Covariance with positive affine transformations (Nash solution), Exercise 309.1
• Crazy players, 239
• Credible objection in coalitional game, 278
• Cycling phase of machine game, 171

D

• Deductive interpretation, 5
• Dictatorial choice rule, 181
• Disagreement outcome in Nash solution, Definition 300.1
• Discounting
  • bargaining game of alternating offers, 119, 122, Example 125.1
  • Nash folk theorem, Proposition 145.2
  • perfect folk theorem, Proposition 151.1
  • preferences, 137
  • repeated game, 139
  • structure of subgame perfect equilibria of repeated game, 153-154
• Dominance solvable, Exercise 63.2
• Dominant action, Example 18.3
• Dominant strategy equilibrium of strategic game, Definition 181.1
• Dominated action, see weakly dominated action, strictly dominated action
• Dove, see Hawk-Dove
• DSE (dominant strategy equilibrium), Definition 181.1
• DUM (Shapley value), 292
• Dummy player in coalitional game, Exercise 280.4
  • axiom of Shapley value, 292
• Dynamic adjustment process, 52

E

• Edgeworth box, 270
• Education, Spence's model, Example 237.2
  • refinement of equilibria, Example 246.1
• Eductive interpretation, 5
• Efficiency in coalitional game, 290
  • Pareto (Nash solution), 305
• Efficient agreement (bargaining), 122, 125
• Electronic mail game, 81-84
• Elimination of dominated actions, see iterated elimination
• Endowment in market, 264
• Enforceable payoff profile/outcome, 143
• Environment (implementation theory), 179
• e-equilibrium, Exercise 108.1
• Equal treatment in core, Lemma 272.2
• Equilibrium, see solution
• Equilibrium, competitive, 266
• Equivalence of extensive games
  • coalescing of moves, 207
  • framing effects, 209
  • inflation-deflation, 205
  • interchange of moves, 208
  • one-player games, Exercise 208.1
  • superfluous moves, 206
• Equivalence of mixed and behavioral strategies, 214
• ESS (evolutionarily stable strategy), 50
• Event, 69
  • self-evident, Definition 73.2
• Evolutionarily stable strategy, Definition 49.1
  • existence, Exercise 51.1
• Evolutionary equilibrium, 48
• Evolutive interpretation, 5
• Excess of a coalition, 283
• Exchange economy, 269
  • core, 271
• Exchange game (Bayesian), Exercise 28.1
• Existence
  • evolutionarily stable strategy, Exercise 51.1
  • mixed strategy equilibrium, Proposition 33.1, 33
  • Nash equilibrium, Proposition 20.3
  • sequential equilibrium, Corollary 253.2
  • subgame perfect equilibrium, Proposition 99.2
  • trembling hand perfect equilibrium, Proposition 249.1, Corollary 253.2
• Exogenous uncertainty in extensive game
  • imperfect information, 200
  • perfect information, 101
• Extensive game, Definition 200.1
  • behavioral strategy, Definition 212.1
  • chance moves, 101, 201
  • equivalence principles
    • addition of superfluous move, 206
    • coalescing of moves, 207
    • inflation-deflation, 205
    • interchange of moves, 208
  • exogenous uncertainty, 101, 200
  • imperfect information, Definition 200.1
  • machine game, Exercise 174.1
  • mixed strategy, Definition 212.1
  • outcome, 213, 223
  • perfect and imperfect recall, Definition 203.3
  • perfect Bayesian equilibrium, Definition 232.1
  • perfect information, Definition 89.1
    • Nash equilibrium, Definition 93.1
    • no indifference condition, Exercise 100.2
    • outcome, 93
    • reduced strategic form, Definition 95.1
    • simultaneous moves, 102
    • strategic form, Definition 94.1
    • strategy, Definition 92.1
    • subgame, Definition 97.1
    • subgame perfect equilibrium, Definition 97.2
  • pure strategy, Definition 203.1
  • sequential equilibrium, see sequential equilibrium
  • simultaneous moves, 202
  • solutions, see solution
  • vs. strategic game, 3
  • trembling hand perfect equilibrium, Definition 251.1
• Extensive game form with perfect information, 90, 179
• External stability in coalitional game, Definition 279.1

F

• Feasible payoff profile
  • coalitional game, 258
  • strategic game, 139
• Fictitious play, 52
• Finite extensive game, 90
• Finite horizon extensive game, 90
  • examples, 105
• Finite strategic game, 11
• Finitely repeated game, 155
  • Nash equilibrium, 155-157
  • Nash folk theorem, Proposition 156.1
  • perfect folk theorem, Proposition 160.1
  • subgame perfect equilibrium, 157-160
• First mover advantage in bargaining game, 126
• First price auction, Exercise 18.2
• Fixed point theorem, Lemma 20.1
• Folk theorem, see Nash folk theorem, perfect folk theorem
• Forget, players who do so, 204
• Forward induction, 110-114
• Framing effects, 209
• Full dimensionality in repeated game, 151

G

• Game form
  • extensive, 201
  • extensive, with perfect information, 90
• Game theory and competitive equilibrium, 3
• Gibbard-Satterthwaite theorem, Proposition 181.2
• Groves mechanism, 184
• Guess the average, Exercise 35.1
  • rationalizability, Exercise 56.5
• Guessing right, Exercise 36.1

H

• Hats, puzzle of, 71
• Hawk-Dove, Example 16.3, Example 49.2
• History in extensive game
  • imperfect information, Definition 200.1
  • perfect information, Definition 89.1
• Homogeneous weighted majority game, Exercise 289.2

I

• IIA (Nash solution axiom), 306
• Impatience in bargaining game, 126
• Imperfect information
  • extensive game, Definition 200.1
  • strategic game, 24
• Imperfect information in game models, 199
• Imperfect recall, 203
• Implementable choice rule, Definition 179.1
• Implementation theory, 177-196
  • dominant strategy implementation, 180-185
  • Nash implementation, 185-191
  • and Nash solution, 311
  • subgame perfect equilibrium, 191-195
  • virtual SPE-implementation, 192
• Imputation, 278
• Increasing function, 7
• Independence of irrelevant alternatives (Nash solution), 306, Exercise 309.1
• Individually rational
  • see enforceable, 143
  • strict (Nash solution), Exercise 309.2
• Indivisible good, market for
  • core, Example 260.1
  • Shapley value, Example 294.4
  • stable set, Exercise 280.2
• Infinitely repeated game, Definition 137.1
  • complexity, 164
  • Nash equilibrium, 143-146
  • subgame perfect equilibrium, 146-154
• Inflation-deflation principle, 205
• Information
  • more is better, Exercise 71.2
  • more may hurt, Exercise 28.2, Exercise 48.1
• Information function, Definition 68.1
  • partitional, Definition 68.2
• Information partition
  • correlated equilibrium, Definition 45.1
  • extensive game with imperfect information, Definition 200.1
• Information set, Definition 200.1
  • interpretation, 205
  • nonordered, Example 223.1, Exercise 229.1
• Initial history, 90
• Interchange of moves
  • equivalence of extensive games, 208
• Interchangeable equilibria
  • extensive game with perfect information, Exercise 100.2
  • strictly competitive game, 23
• Interchangeable players in coalitional game, 292
• Internal stability in coalitional game, Definition 279.1
• Interpretation
  • forward induction, 112
  • information set, 205
  • mixed strategy Nash equilibrium, 37-44
  • solutions, 5
  • state of the world, 67
  • strategic game, 13
  • strategy in extensive game, 103
• Introductory phase of machine game, 171
• Investment race, Exercise 35.2
• Irrational players, 239
• Irrelevant alternatives, independence of, 306
• Iterated elimination
  • forward induction, 110-114
  • and rationalizable actions, Proposition 61.2
  • strictly dominated actions, Definition 60.2
  • subgame perfect equilibrium, 108
  • weakly dominated actions, 62
    • and subgame perfect equilibrium, 108

J

• Judgment of Solomon, see Solomon's predicament

K

• Kakutani's fixed point theorem, Lemma 20.1
• Kalai-Smorodinsky solution, 310
• Kernel of coalitional game, Definition 284.1
  • and bargaining set, Lemma 285.1
  • and nucleolus, Lemma 287.1
• Knowledge
  • common, Definition 73.1, Definition 73.2
  • mutual, 73
• Knowledge and solution concepts, 76-81
• Knowledge function, 69, 70
• Kuhn's theorem, Proposition 99.2
  • in game with chance moves, Exercise 102.1
  • in game with simultaneous moves, Exercise 103.3

L

• Leader-follower game, Example 97.3
• Learning, 52
• Lexicographic minimality in nucleolus of coalitional game, Lemma 286.2
• Lexicographic preferences in machine game, 165, 172-174
• Limit of means
  • Nash folk theorem, Proposition 144.3
  • perfect folk theorem, Proposition 146.2
  • preferences, 138
  • repeated game, 139
• Location game, Example 18.6
  • iterated elimination of dominated actions, Exercise 63.1
  • rationalizable actions, Exercise 57.1
• Long- and short-lived players in infinite game, Exercise 148.1

M

• Machine, 140, 164
  • complexity, 165
• Machine game, Definition 165.1
  • cycling/introductory phases, 171
  • extensive game and complexity, Exercise 174.1
  • lexicographic preferences, 165
  • structure of equilibria, 168-174
• Majority game, Example 295.3, Exercise 295.4
  • bargaining set, Example 282.2
  • core, Example 259.1, Example 260.3
  • kernel, Example 285.2
  • stable set, Example 279.3
  • weighted, Exercise 289.2
    • nucleolus, Exercise 289.2
    • Shapley value, Example 294.1
• Marginal contribution of player in coalitional game, 291
• Market for indivisible good
  • core, Example 260.1
  • Shapley value, Example 294.4
  • stable set, Exercise 280.2
• Market with transferable payoff, 263
  • bargaining set, Exercise 283.1
  • core, 264
    • nonempty, Proposition 264.2
• Markovian machine, 143
• Matching pennies, Example 17.1
• Maxminimizer, Definition 21.2, 209
• Mechanism design, see implementation theory
• Memory, poor (extensive games that model), 204
• Message
  • in Beer or Quiche game, Example 244.2
  • in signaling game, 237
• Minmax payoff, 143
• Mistakes, 247
• Mixed extension
  • strategic game, Definition 32.1
  • strictly competitive, 36
• Mixed strategy
  • as belief, 43
  • extensive game
    • imperfect information, Definition 212.1
    • perfect information, 93
  • naïve interpretation, 37
  • outcome equivalence with behavioral strategy, 214
  • as pure strategy in extended game, 39
  • strategic game, 32
• Mixed strategy equilibrium
  • approachable, 42
  • and correlated equilibrium, Proposition 45.3
  • extensive game, 216
  • Harsanyi's model, 41-43
  • interpretation, 37-44
  • knowledge requirements, Proposition 78.1
  • as steady state, 38
  • strategic game, Definition 32.3, Definition 44.1
• Monotonic choice rule, Definition 186.1
• Mutation, 49
• Mutual knowledge, 73
• My aunt and I (coalitional game)
  • bargaining set, Example 282.3
  • generalization, Example 295.3
  • kernel, Example 285.3

N

• Naïve interpretation of mixed strategy, 37
• Nash bargaining
  • and bargaining game of alternating offers, 310-311
  • comparative statics of risk aversion, Proposition 304.1
  • independence of irrelevant alternatives, 306
  • Pareto efficiency, 305
  • problem, Definition 300.1
  • symmetry, Definition 305.1
• Nash equilibrium
  • Bayesian game, Definition 26.1
  • behavioral strategies, 216
  • and correlated equilibrium, Proposition 45.3
  • existence for strategic game, Proposition 20.3
  • extensive game with perfect information, Definition 93.1
  • finitely repeated game, 155-157
  • infinitely repeated game, 143-146
  • interchangeability, 23
  • knowledge requirements, Proposition 77.1
  • mixed strategy of extensive game, 216
  • mixed strategy of strategic game, Definition 32.3, 43
  • strategic game, Definition 14.1
    • and maxminimizers, Proposition 22.2
  • and trembling hand perfect equilibrium, 248
• Nash folk theorem
  • discounting criterion, Proposition 145.2
  • finitely repeated game, Proposition 156.1
  • limit of means criterion, Proposition 144.3
• Nash implementation, 185-191
  • revelation principle, Lemma 185.2
• Nash solution, Definition 301.2
  • axiomatic definition, 305-310
  • exact implementation, 311
• Nature, moves of, see chance moves
• Negotiation, see bargaining game of alternating offers, Nash bargaining
• Never-best response, Definition 59.1
• No indifference condition, Exercise 100.2
• No veto power, Definition 187.1
• Non-redundancy in Nash bargaining problem, Definition 300.1
• Noncooperative vs. coalitional games, 2, 255
• Nondecreasing function, 7
• Nonempty core
  • coalitional game, Proposition 262.1
  • market with transferable payoff, Proposition 264.2
• Nonordered information sets, Example 223.1, Exercise 229.1
• Nontransferable payoff coalitional game, Definition 268.2
• Normal form game, see strategic game
• Nucleolus of coalitional game, Definition 286.1
  • and kernel, Lemma 287.1

O

• Objection in coalitional game
  • bargaining set, 281
  • kernel, 284
  • Nash solution, 302
  • nucleolus, 285
  • Shapley value, 290
  • stable set, 278
• One deviation property
  • sequential equilibrium, Exercise 227.1
  • subgame perfect equilibrium, Lemma 98.2
    • bargaining game, Exercise 123.1
    • game with chance moves, Exercise 102.1
    • game with simultaneous moves, Exercise 103.3
    • infinitely repeated game with discounting, Lemma 153.1
• Opting out (bargaining), 128
• Outcome
  • extensive game
    • imperfect information, 213, 223
    • perfect information, 93
  • implementation theory, 178
  • strategic game, 11
• Outcome-equivalence of mixed and behavioral strategies, 214
• Outcome-equivalent strategies, 94
• Output function of machine, 140, 164
• Outside option
  • in bargaining, 128
  • in BoS, Example 110.1
• Outside option principle, 129
• Overtaking
  • perfect folk theorem, Proposition 149.1
  • preferences, 139
  • repeated game, 139

P

• PAR (Nash solution axiom), 305
• Paradoxes in finite horizon games, 105
• Paratroopers, 170
• Pareto efficiency, 7
  • axiom of Nash solution, 305
• Pareto frontier of agreement set, 122
• Parliament, coalitional game model, Example 295.3, Exercise 295.4
• Partition of a set, 7
• Partitional information function, Definition 68.2
• Payoff function, 13
• Payoff profile
  • repeated game
    • discounting, 138
    • limit of means, 138
  • strategic game, 139
• Perfect Bayesian equilibrium, Definition 232.1
  • and sequential equilibrium, Proposition 234.1
• Perfect folk theorem
  • discounting criterion, Proposition 151.1
  • finitely repeated game, Proposition 160.1
  • limit of means criterion, Proposition 146.2
  • overtaking criterion, Proposition 149.1
• Perfect information extensive game, Definition 89.1
  • strategic form, Definition 94.1
• Perfect recall, Definition 203.3
• Perturbed game
  • chain-store, Example 239.1
  • mixed and pure strategies, 42
  • trembling hand perfect equilibrium, 247
• Phases of equilibrium of machine game, 171
• f, see Shapley value
• Plan of action vs. strategy, 103
• Planner (implementation theory), 177
• Player function in extensive game
  • imperfect information, Definition 200.1
  • perfect information, Definition 89.1
• Pollute the lake, Exercise 261.3
• Pooling equilibrium in Spence's model, Example 237.2
  • eliminated by refinement, Example 246.1
• Pre-trial negotiation, Exercise 246.2
• Preference relation, 7
  • repeated game, 137
• Prior belief
  • in Bayesian game, Definition 25.1
  • and posterior belief, 75
• Prisoner's dilemma, Example 16.2
  • equilibria of machine game, Example 166.2, 169, 172
  • grim strategy in repeated game, Example 141.1
    • complexity, Example 166.1
  • infinitely vs. finitely repeated game, 134
  • modified in finitely repeated game, 159
  • rationalizability, 56
• Probability measure, 7
• Production economy
  • core, Exercise 259.3
  • core convergence, Exercise 268.1
  • nucleolus, Exercise 289.1
  • Shapley value, Exercise 295.2
• Production function, 264
• Profile, 7
• Proper equilibrium, 254
• Punishment
  • limited length of time, Proposition 146.2
  • punishing the punisher, Proposition 149.1
  • rewarding players who punish, 150, 159
  • trigger strategies, 143
• Pure strategy
  • extensive game, Definition 203.1
  • strategic game, 32
• Purification of mixed strategy equilibrium, 39
• Puzzle of the hats, 71

Q

• Quasi-concave preference relation, 7, 20

R

• Rational choice, 4
• Rational, individually, see enforceable, 143
• Rationalizability, 53
  • independence vs. correlation in beliefs, 57
  • and iterated elimination of strictly dominated actions, Proposition 61.2
  • knowledge requirements, Proposition 80.1
• Rationalizable action, Definition 54.1, Definition 55.1
• Recall, 203
• Reduced strategic form, Definition 95.1
• Reduced strategy, 94
• Refinements of sequential equilibrium, 243-246
• Relative probabilities, 254
• Renegotiation, 161
• Repeated game
  • constituent game, 136
  • finite, 155
  • finite vs. infinite, 134
  • forms of preference relation, 137-139
  • infinite, Definition 137.1
  • see also finitely repeated game, infinitely repeated game
• Reputation, 238-243
• Restrictions on beliefs, 243-246
• Revelation principle
  • DSE implementation, Lemma 181.4
  • Nash implementation, Lemma 185.2
• Reversal of beliefs, Example 236.1
• Risk of breakdown in bargaining, 129
  • and Nash solution, Proposition 310.3
• Risk, comparative statics (Nash bargaining), Proposition 304.1

S

• S see coalition
• S-feasible payoff vector, 258
• Second price auction, Exercise 18.3
  • imperfect information, Example 27.1
• Self-evident event, Definition 73.2
• Selten's horse
  • sequential equilibria, Example 225.2
  • trembling hand perfect equilibria, Example 252.1
• Separating equilibrium, Example 237.2, Example 246.1
• Sequential equilibrium, Definition 225.1
  • assessment, Definition 222.1
  • belief system, 223
  • coalescing of moves, Example 226.2
  • existence, Corollary 253.2
  • and perfect Bayesian equilibrium, Proposition 234.1
  • refinements, 243-246
  • restrictions on beliefs, 243-246
  • reversal of beliefs, Example 236.1
  • and structural consistency, 228-231
  • and trembling hand perfect equilibrium, Proposition 251.2
• Sequential rationality, 221, 223, Definition 224.1
• Shapley value, Definition 291.2
  • axiomatic characterization, 292
  • objections and counterobjections, 289
• Short-lived players in infinite game, Exercise 148.1
• Shouting game in implementation, 188, 189
• Signal function in Bayesian game, Definition 25.1
• Signaling game, 237
  • pooling equilibrium, Example 237.2
    • eliminated by refinement, Example 246.1
  • separating equilibrium, Example 237.2, Example 246.1
• Simple coalitional game, Exercise 261.1
  • core, Exercise 261.1
  • stable sets, Exercise 280.1
• Simultaneous moves in extensive game
  • imperfect information, 202
  • perfect information, 102
• Solomon's predicament, Example 186.3, Example 190.1, Example 191.2
• Solution
  • for coalitional games
    • bargaining set, Definition 282.1
    • core (nontransferable payoff), Definition 268.3
    • core (transferable payoff), Definition 258.2
    • general idea, 255
    • kernel, Definition 284.1
    • Nash solution, Definition 301.2
    • nucleolus, Definition 286.1
    • Shapley value, Definition 291.2
    • stable sets, Definition 279.1
  • deductive interpretation, 5
  • for extensive games
    • perfect Bayesian equilibrium, Definition 232.1
    • sequential equilibrium, Definition 225.1
    • subgame perfect equilibrium, Definition 97.2
    • trembling hand perfect equilibrium, Definition 251.1
  • general idea, 2
  • implementation theory, 179
  • steady state interpretation, 5
  • for strategic games
    • correlated equilibrium, Definition 45.1
    • dominant strategy equilibrium, Definition 181.1
    • evolutionary equilibrium, Definition 49.1
    • iterated elimination of strictly dominated actions, Definition 60.2
    • mixed strategy equilibrium, Definition 32.3
    • Nash equilibrium, Definition 14.1
    • rationalizability, Definition 54.1
    • trembling hand perfect equilibrium, Definition 248.1
• SPE, see subgame perfect equilibrium
• Spence's model of education, Example 237.2
  • refinement of equilibria, Example 246.1
• Split-the-pie game, Example 120.1
  • subgame perfect equilibrium, Example 125.1
• Stable set of coalitional game, Definition 279.1
  • and core, Proposition 279.2
  • generalized, Exercise 280.5
• Stackelberg game, Example 97.3
• Standard of behavior in coalitional game, 279
• State of machine, 140, 164
• State of the world, 67
• Stationarity of strategies in bargaining game, 126
• Steady state interpretation, 5, 14
• Strategic form
  • agent strategic form, 250
  • extensive game with perfect information, Definition 94.1
• Strategic game, Definition 11.1
  • Bayesian, Definition 25.1
  • dominant strategy equilibrium, Definition 181.1
  • examples, 15-19
  • existence of Nash equilibrium, Proposition 20.3
  • vs. extensive game, 3
  • form of extensive game, Definition 94.1
  • imperfect information, 24
  • interpretation, 13
  • mixed extension, Definition 32.1
  • as reduced strategic form, Definition 95.1
  • solutions, see solution
  • strictly competitive, Definition 21.1
  • symmetric, Exercise 20.4
  • tabular representation, 13
  • trembling hand perfect equilibrium, Definition 248.1
• Strategic game form, 178
• Strategy
  • equilibrium structure in repeated game, 134, 163
  • extensive game
    • imperfect information, Definition 203.1
    • interpretation, 103
    • perfect information, Definition 92.1
    • perfect information and chance moves, 102
    • perfect information and simultaneous moves, 102
    • vs. plan of action, 103
  • as machine in repeated game, 140
  • stationarity in bargaining game, 126
• Strict equilibrium, 50
• Strict individual rationality
  • Nash bargaining, Exercise 309.2
• Strict individual rationality (strict enforceability), 143
• Strictly competitive strategic game, Example 17.1, Definition 21.1
  • value, 23
• Strictly dominated action, Definition 59.2
• Strictly enforceable payoff profile/outcome, 143
• Structural consistency, 222, Definition 228.1
  • and sequential equilibrium, 228-231
• Structure of equilibria of repeated game, 153-154
  • complexity, 163
  • cycling/introductory phases, 171
• Subgame of extensive game with perfect information, Definition 97.1
• Subgame perfect equilibrium
  • extensive game with imperfect information, see sequential equilibrium
  • extensive game with perfect information, Definition 97.2
  • finitely repeated game, Proposition 157.2, Proposition 160.1
  • implementation, 191
    • Nash solution, 311
  • infinitely repeated game, 146-154
    • discounting, Proposition 151.1, Proposition 154.1
    • limit of means, Proposition 146.2
    • overtaking, Proposition 149.1
  • interchangeability, Exercise 100.2
  • iterated elimination of weakly dominated actions, 108
  • virtual SPE-implementation, 192
• Superadditivity of coalitional game, 258
• Superfluous moves
  • equivalence of extensive games, 206
• Support of probability distribution, 32
• SYM (Nash solution axiom), 305
• SYM (Shapley value axiom), 292
• Symmetric bargaining problem, Definition 305.1
• Symmetric game, Exercise 20.4
  • evolutionary equilibrium, Exercise 51.1
• Symmetry axiom
  • Nash solution, 305
  • Shapley value, 292
• Symmetry function, Definition 305.1

T

• Terminal history in extensive game
  • imperfect information, Definition 200.1
  • perfect information, Definition 89.1
• Three-player bargaining, 130
• Three-player majority game
  • bargaining set, Example 282.2
  • core, Example 259.1
  • kernel, Example 285.2
  • stable set, Example 279.3
• Transferable payoff
  • coalitional game, Definition 257.1
  • coalitional game without, Definition 268.2
  • core, Definition 258.2
  • market, 263
• Transition function of machine, 140, 164
• Treasure in the mountains, Example 259.2
• Trembling hand perfect equilibrium, 246-253
  • extensive game, Definition 251.1
    • existence, Corollary 253.2
    • and sequential equilibrium, Proposition 251.2
    • and weakly dominated strategy, 252
  • strategic game, Definition 248.1
    • existence, Proposition 249.1
    • and Nash equilibrium, 248
    • and weakly dominated strategy, Proposition 248.2
• Trigger strategy in repeated game, 143
• Truthful implementation, Definition 179.2
  • and DSE-implementation, Lemma 181.4
  • and Nash implementation, Lemma 185.2
  • via Groves mechanism, Proposition 184.2
• Type of agent in exchange economy, 272
• Types of players, 24, Definition 231.1

U

• Uncertainty
  • in extensive game
    • imperfect information, 200
    • perfect information, 101
  • in model of rational choice, 4, Exercise 71.2
  • about others' characteristics, 24
  • about others' knowledge, 29
• Unique best agreements in Nash bargaining problem, Definition 300.1
• Utility function, 4
  • see also payoff function, 13

V

• Value
  • coalitional game, 290
  • strictly competitive game, 23
• Veto player in coalitional game, Exercise 261.1
• Veto power, Definition 187.1
• Virtual SPE-implementation, 192
• Von Neumann-Morgenstern utility function, 5

W

• War of attrition, Example 18.4
• Weak separability of preferences in repeated game, Definition 137.1
• Weakly dominant action, Exercise 18.3
• Weakly dominated action, Definition 62.1
  • and trembling hand perfect equilibrium, Proposition 248.2
• Weighted majority game, Exercise 289.2
  • Shapley value, Example 294.1
• Winning coalition, Exercise 261.1
• Worth of a coalition, Definition 257.1

Z

• Zerosum coalitional game, Exercise 261.2
  • weighted majority game, Exercise 289.2
• Zerosum strategic game, 21