A course in game theory
by Martin J. Osborne and Ariel Rubinstein
Index
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