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Introduction to Epistemic Game Theory
Eric Pacuit
University of Maryland, College Parkpacuit.org
May 25, 2016
Eric Pacuit 1
Plan
I Introductory remarksI Games and Game ModelsI Backward and Forward Induction ReasoningI Concluding remarks
Eric Pacuit 2
The Guessing Game
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
dev.pacuit.org/games/avg
Eric Pacuit 3
The Guessing Game, again
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
dev.pacuit.org/games/avg
Eric Pacuit 4
The Guessing Game, again
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
dev.pacuit.org/games/avg
Eric Pacuit 4
Traveler’s Dilemma
1. You and your friend write down an integer between 2 and 100(without discussing).
2. If both of you write down the same number, then both will receivethat amount in Euros from the airline in compensation.
3. If the numbers are different, then the airline assumes that thesmaller number is the actual price of the luggage.
4. The person that wrote the smaller number will receive that amountplus 2 EUR (as a reward), and the person that wrote the largernumber will receive the smaller number minus 2 EUR (as apunishment).
Suppose that you are randomly paired with another person here atNASSLLI. What number would you write down?
dev.pacuit.org/games/td
Eric Pacuit 5
Traveler’s Dilemma
1. You and your friend write down an integer between 2 and 100(without discussing).
2. If both of you write down the same number, then both will receivethat amount in Euros from the airline in compensation.
3. If the numbers are different, then the airline assumes that thesmaller number is the actual price of the luggage.
4. The person that wrote the smaller number will receive that amountplus 2 EUR (as a reward), and the person that wrote the largernumber will receive the smaller number minus 2 EUR (as apunishment).
Suppose that you are randomly paired with another person here atNASSLLI. What number would you write down?
dev.pacuit.org/games/td
Eric Pacuit 5
Traveler’s Dilemma
1. You and your friend write down an integer between 2 and 100(without discussing).
2. If both of you write down the same number, then both will receivethat amount in Euros from the airline in compensation.
3. If the numbers are different, then the airline assumes that thesmaller number is the actual price of the luggage.
4. The person that wrote the smaller number will receive that amountplus 2 EUR (as a reward), and the person that wrote the largernumber will receive the smaller number minus 2 EUR (as apunishment).
Suppose that you are randomly paired with another person here atNASSLLI. What number would you write down?
dev.pacuit.org/games/td
Eric Pacuit 5
Traveler’s Dilemma
1. You and your friend write down an integer between 2 and 100(without discussing).
2. If both of you write down the same number, then both will receivethat amount in Euros from the airline in compensation.
3. If the numbers are different, then the airline assumes that thesmaller number is the actual price of the luggage.
4. The person that wrote the smaller number will receive that amountplus 2 EUR (as a reward), and the person that wrote the largernumber will receive the smaller number minus 2 EUR (as apunishment).
Suppose that you are randomly paired with another person here atNASSLLI. What number would you write down?
dev.pacuit.org/games/td
Eric Pacuit 5
Traveler’s Dilemma
1. You and your friend write down an integer between 2 and 100(without discussing).
2. If both of you write down the same number, then both will receivethat amount in Euros from the airline in compensation.
3. If the numbers are different, then the airline assumes that thesmaller number is the actual price of the luggage.
4. The person that wrote the smaller number will receive that amountplus 2 EUR (as a reward), and the person that wrote the largernumber will receive the smaller number minus 2 EUR (as apunishment).
Suppose that you are randomly paired with another person here atNASSLLI. What number would you write down?
dev.pacuit.org/games/td
Eric Pacuit 5
From Decisions to Games, I
Commenting on the difference between Robin Crusoe’s maximizationproblem and the maximization problem faced by participants in a socialeconomy, von Neumann and Morgenstern write:
“Every participant can determine the variables which describe his ownactions but not those of the others. Nevertheless those “alien”variables cannot, from his point of view, be described by statisticalassumptions.
This is because the others are guided, just as he himself,by rational principles—whatever that may mean—and no modusprocedendi can be correct which does not attempt to understand thoseprinciples and the interactions of the conflicting interests of allparticipants.”addasdfafds (vNM, pg. 11)
Eric Pacuit 6
From Decisions to Games, I
Commenting on the difference between Robin Crusoe’s maximizationproblem and the maximization problem faced by participants in a socialeconomy, von Neumann and Morgenstern write:
“Every participant can determine the variables which describe his ownactions but not those of the others. Nevertheless those “alien”variables cannot, from his point of view, be described by statisticalassumptions. This is because the others are guided, just as he himself,by rational principles—whatever that may mean—and no modusprocedendi can be correct which does not attempt to understand thoseprinciples and the interactions of the conflicting interests of allparticipants.”addasdfafds (vNM, pg. 11)
Eric Pacuit 6
Just Enough Game Theory
A game is a mathematical model of a social interaction that includes
I the players (N);I the actions (strategies) the players can take (for i ∈ N, Si , ∅);I the players’ interests (for i ∈ N, ui : Πi∈NSi → R); andI the “structure” of the decision problem.
It does not specify the actions that the players do take.
Eric Pacuit 7
Just Enough Game Theory
A game is a mathematical model of a social interaction that includes
I the players (N);I the actions (strategies) the players can take (for i ∈ N, Si , ∅);I the players’ interests (for i ∈ N, ui : Πi∈NSi → R); andI the “structure” of the decision problem.
It does not specify the actions that the players do take.
Eric Pacuit 7
Bl r
Au 3, 1 0, 0d 0, 0 1, 3
A
B B
(3,1) (0,0) (0,0) (1,3)
u d
l r l r
Bl r
Au 3, 1 0, 0d 0, 0 1, 3
A
2, 2In
Out
Eric Pacuit 8
Bl r
Au 3, 1 0, 0d 0, 0 1, 3
A
B B
(3,1) (0,0) (0,0) (1,3)
u d
l r l r
Bl r
Au 3, 1 0, 0d 0, 0 1, 3
A
2, 2In
Out
Eric Pacuit 8
Bl r
Au 3, 1 0, 0d 0, 0 1, 3
A
B B
(3,1) (0,0) (0,0) (1,3)
u d
l r l r
Bl r
Au 3, 1 0, 0d 0, 0 1, 3
A
2, 2In
Out
Eric Pacuit 8
Bl r
Au 3, 1 0, 0d 0, 0 1, 3
A
B B
(3,1) (0,0) (0,0) (1,3)
u d
l r l r
Bl r
Au 3, 1 0, 0d 0, 0 1, 3
A
2, 2In
Out
Eric Pacuit 8
Just Enough Game Theory: Solution Concepts
A solution concept is a systematic description of the outcomes thatmay emerge in a family of games.
This is the starting point for most of game theory and includes manyvariants: Nash equilibrium (and refinements), backwards induction, oriterated dominance of various kinds.
They are usually thought of as the embodiment of “rational behavior” insome way and used to analyze game situations.
Eric Pacuit 9
Game Models
I A game is a partial description of a set (or sequence) ofinterdependent (Bayesian) decision problems.
A game will not normally contain enough information to determinewhat the players believe about each other.
I A model of a game is a completion of the partial specification of theBayesian decision problems and a representation of a particularplay of the game.
I There are no special rules of rationality telling one what to do in theabsence of degrees of belief except: decide what you believe, andthen maximize (subjective) expected utility.
Eric Pacuit 10
Game Models
I A game is a partial description of a set (or sequence) ofinterdependent (Bayesian) decision problems.
A game will not normally contain enough information to determinewhat the players believe about each other.
I A model of a game is a completion of the partial specification of theBayesian decision problems and a representation of a particularplay of the game.
I There are no special rules of rationality telling one what to do in theabsence of degrees of belief except: decide what you believe, andthen maximize (subjective) expected utility.
Eric Pacuit 10
Game Models
I A game is a partial description of a set (or sequence) ofinterdependent (Bayesian) decision problems.
A game will not normally contain enough information to determinewhat the players believe about each other.
I A model of a game is a completion of the partial specification of theBayesian decision problems and a representation of a particularplay of the game.
I There are no special rules of rationality telling one what to do in theabsence of degrees of belief except: decide what you believe, andthen maximize (subjective) expected utility.
Eric Pacuit 10
Game Models
I A game is a partial description of a set (or sequence) ofinterdependent (Bayesian) decision problems.
A game will not normally contain enough information to determinewhat the players believe about each other.
I A model of a game is a completion of the partial specification of theBayesian decision problems and a representation of a particularplay of the game.
I There are no special rules of rationality telling one what to do in theabsence of degrees of belief except: decide what you believe, andthen maximize (subjective) expected utility.
Eric Pacuit 10
Knowledge and beliefs in game situations
J. Harsanyi. Games with incomplete information played by “Bayesian” players I-III. Man-agement Science Theory 14: 159-182, 1967-68.
R. Aumann. Interactive Epistemology I & II. International Journal of Game Theory (1999).
P. Battigalli and G. Bonanno. Recent results on belief, knowledge and the epistemic foun-dations of game theory. Research in Economics (1999).
R. Myerson. Harsanyi’s Games with Incomplete Information. Special 50th anniversaryissue of Management Science, 2004.
Eric Pacuit 11
John C. Harsanyi, nobel prize winner in economics, developed a theoryof games with incomplete information.
1. incomplete information: uncertainty about the structure of the game(outcomes, payoffs, strategy space)
2. imperfect information: uncertainty within the game about theprevious moves of the players
J. Harsanyi. Games with incomplete information played by “Bayesian” players I-III. Man-agement Science Theory 14: 159-182, 1967-68.
Eric Pacuit 12
John C. Harsanyi, nobel prize winner in economics, developed a theoryof games with incomplete information.
1. incomplete information: uncertainty about the structure of the game(outcomes, payoffs, strategy space)
2. imperfect information: uncertainty within the game about theprevious moves of the players
J. Harsanyi. Games with incomplete information played by “Bayesian” players I-III. Man-agement Science Theory 14: 159-182, 1967-68.
Eric Pacuit 12
Information in games situations
I Various states of information disclosure.
• ex ante, ex interim, ex post
I Various “types” of information:• imperfect information about the play of the game• incomplete information about the structure of the game• strategic information (what will the other players do?)• higher-order information (what are the other players thinking?)
I Varieties of informational attitudes• hard (“knowledge”)• soft (“beliefs”)
Eric Pacuit 13
Information in games situations
I Various states of information disclosure.• ex ante, ex interim, ex post
I Various “types” of information:• imperfect information about the play of the game• incomplete information about the structure of the game• strategic information (what will the other players do?)• higher-order information (what are the other players thinking?)
I Varieties of informational attitudes• hard (“knowledge”)• soft (“beliefs”)
Eric Pacuit 13
Information in games situations
I Various states of information disclosure.• ex ante, ex interim, ex post
I Various “types” of information:
• imperfect information about the play of the game• incomplete information about the structure of the game• strategic information (what will the other players do?)• higher-order information (what are the other players thinking?)
I Varieties of informational attitudes• hard (“knowledge”)• soft (“beliefs”)
Eric Pacuit 13
Information in games situations
I Various states of information disclosure.• ex ante, ex interim, ex post
I Various “types” of information:• imperfect information about the play of the game• incomplete information about the structure of the game• strategic information (what will the other players do?)• higher-order information (what are the other players thinking?)
I Varieties of informational attitudes• hard (“knowledge”)• soft (“beliefs”)
Eric Pacuit 13
Information in games situations
I Various states of information disclosure.• ex ante, ex interim, ex post
I Various “types” of information:• imperfect information about the play of the game• incomplete information about the structure of the game• strategic information (what will the other players do?)• higher-order information (what are the other players thinking?)
I Varieties of informational attitudes
• hard (“knowledge”)• soft (“beliefs”)
Eric Pacuit 13
Information in games situations
I Various states of information disclosure.• ex ante, ex interim, ex post
I Various “types” of information:• imperfect information about the play of the game• incomplete information about the structure of the game• strategic information (what will the other players do?)• higher-order information (what are the other players thinking?)
I Varieties of informational attitudes• hard (“knowledge”)• soft (“beliefs”)
Eric Pacuit 13
Epistemic Game Theory
Formally, a game is described by its strategy sets andpayoff functions.
But in real life, may other parametersare relevant; there is a lot more going on. Situationsthat substantively are vastly different may neverthelesscorrespond to precisely the same strategic game....
The difference lies in the attitudes of the players, in theirexpectations about each other, in custom, and in his-tory, though the rules of the game do not distinguishbetween the two situations. (pg. 72)
R. Aumann and J. H. Dreze. Rational Expectations in Games. American Economic Re-view 98 (2008), pp. 72-86.
Eric Pacuit 14
Epistemic Game Theory
Formally, a game is described by its strategy sets andpayoff functions. But in real life, may other parametersare relevant; there is a lot more going on. Situationsthat substantively are vastly different may neverthelesscorrespond to precisely the same strategic game.
...
The difference lies in the attitudes of the players, in theirexpectations about each other, in custom, and in his-tory, though the rules of the game do not distinguishbetween the two situations. (pg. 72)
R. Aumann and J. H. Dreze. Rational Expectations in Games. American Economic Re-view 98 (2008), pp. 72-86.
Eric Pacuit 14
Epistemic Game Theory
Formally, a game is described by its strategy sets andpayoff functions. But in real life, may other parametersare relevant; there is a lot more going on. Situationsthat substantively are vastly different may neverthelesscorrespond to precisely the same strategic game....
The difference lies in the attitudes of the players, in theirexpectations about each other, in custom, and in his-tory, though the rules of the game do not distinguishbetween the two situations. (pg. 72)
R. Aumann and J. H. Dreze. Rational Expectations in Games. American Economic Re-view 98 (2008), pp. 72-86.
Eric Pacuit 14
The Epistemic Program in Game Theory
...the analysis constitutes a fleshing-out of the textbook interpretation ofequilibrium as ‘rationality plus correct beliefs.’
E. Dekel and M. Siniscalchi. Epistemic Game Theory. Handbook of Game Theory withEconomic Applications, 2015.
A. Brandenburger. The Language of Game Theory: Putting Epistemics into the Mathe-matics of Games. World Scientific, 2014.
A. Perea. Epistemic Game Theory: Reasoning and Choice. Cambridge University Press,2012.
EP and O. Roy. Epistemic Game Theory. Stanford Encyclopedia of Philosophy, 2015.
Eric Pacuit 15
Bayesian Decision Problem
it rains it does not rain
take umbrella encumbered, dry encumbered, dry
leave umbrella wet free, dry
States: it rains; it does not rain
Outcomes: encumbered, dry; wet; free, dry
Actions: take umbrella; leave umbrella
Eric Pacuit 16
EU(A) =∑
o∈O PA(o) × U(o)
Expected utility of action A Utility of outcome o
Probability of outcome o conditional on A
Eric Pacuit 17
EU(A) =∑
o∈O PA(o) × U(o)
Expected utility of action A Utility of outcome o
Probability of outcome o conditional on A
Eric Pacuit 17
EU(A) =∑
o∈O PA(o) × U(o)
Expected utility of action A Utility of outcome o
Probability of outcome o conditional on A
Eric Pacuit 17
EU(A) =∑
o∈O PA(o) × U(o)
Expected utility of action A Utility of outcome o
Probability of outcome o conditional on A
Eric Pacuit 17
PA (o): probability of o conditional on A — how likely it is that outcome owill occur, on the supposition that the agent chooses act A .
Evidential: PA (o) = P(o | A) =P(o & A)
P(A)
Classical: PA (o) =∑
s∈S P(s)fA ,s(o), where
fA ,s(o) =
1 A(s) = o
0 A(s) , o
Causal: PA (o) = P(A � o)
P(“if A were performed, outcome o would ensue”)
(Lewis, 1981)
Eric Pacuit 18
PA (o): probability of o conditional on A — how likely it is that outcome owill occur, on the supposition that the agent chooses act A .
Evidential: PA (o) = P(o | A) =P(o & A)
P(A)
Classical: PA (o) =∑
s∈S P(s)fA ,s(o), where
fA ,s(o) =
1 A(s) = o
0 A(s) , o
Causal: PA (o) = P(A � o)
P(“if A were performed, outcome o would ensue”)
(Lewis, 1981)
Eric Pacuit 18
PA (o): probability of o conditional on A — how likely it is that outcome owill occur, on the supposition that the agent chooses act A .
Evidential: PA (o) = P(o | A) =P(o & A)
P(A)
Classical: PA (o) =∑
s∈S P(s)fA ,s(o), where
fA ,s(o) =
1 A(s) = o
0 A(s) , o
Causal: PA (o) = P(A � o)
P(“if A were performed, outcome o would ensue”)
(Lewis, 1981)
Eric Pacuit 18
PA (o): probability of o conditional on A — how likely it is that outcome owill occur, on the supposition that the agent chooses act A .
Evidential: PA (o) = P(o | A) =P(o & A)
P(A)
Classical: PA (o) =∑
s∈S P(s)fA ,s(o), where
fA ,s(o) =
1 A(s) = o
0 A(s) , o
Causal: PA (o) = P(A � o)
P(“if A were performed, outcome o would ensue”)
(Lewis, 1981)
Eric Pacuit 18
B
A
U L R
U 2,1 0,0 U
D 0,0 1,2 U
A
U PlayB(L) PlayB(R)
U 2 0 U
D 0 1 UB
U PlayA(U) PlayA(D)
L 1 0 L
R 0 2 R
Eric Pacuit 19
Models of Games
Suppose that G is a game.
I Outcomes of the game: S = Πi∈NSi
I Player i’s partial beliefs (or conjecture): Pi ∈ ∆(S−i)
∆(X) is the set of probabilities measures over X
Eric Pacuit 20
Models of Games, continued
G = 〈N, {Si , ui}i∈N〉 is a strategic (form of a) game.
I W is a set of possible worlds (possible outcomes of the game)
I s is a function s : W → Πi∈NSi
I For si ∈ Si , [si] = {w | s(w) = si}, if X ⊆ S, [X ] =⋃
s∈X [s].
I ex ante beliefs: For each i ∈ N, Pi ∈ ∆(W). Two assumptions:• [s] is measurable for all strategy profiles s ∈ S• Pi([si]) > 0 for all si ∈ Si
Eric Pacuit 21
Models of Games, continued
G = 〈N, {Si , ui}i∈N〉 is a strategic (form of a) game.
I W is a set of possible worlds (possible outcomes of the game)
I s is a function s : W → Πi∈NSi
I For si ∈ Si , [si] = {w | s(w) = si}, if X ⊆ S, [X ] =⋃
s∈X [s].
I ex ante beliefs: For each i ∈ N, Pi ∈ ∆(W). Two assumptions:• [s] is measurable for all strategy profiles s ∈ S• Pi([si]) > 0 for all si ∈ Si
Eric Pacuit 21
Models of Games, continued
G = 〈N, {Si , ui}i∈N〉 is a strategic (form of a) game.
I W is a set of possible worlds (possible outcomes of the game)
I s is a function s : W → Πi∈NSi
I For si ∈ Si , [si] = {w | s(w) = si}, if X ⊆ S, [X ] =⋃
s∈X [s].
I ex ante beliefs: For each i ∈ N, Pi ∈ ∆(W). Two assumptions:• [s] is measurable for all strategy profiles s ∈ S• Pi([si]) > 0 for all si ∈ Si
Eric Pacuit 21
Models of Games, continued
G = 〈N, {Si , ui}i∈N〉 is a strategic (form of a) game.
I W is a set of possible worlds (possible outcomes of the game)
I s is a function s : W → Πi∈NSi
I For si ∈ Si , [si] = {w | s(w) = si}, if X ⊆ S, [X ] =⋃
s∈X [s].
I ex ante beliefs: For each i ∈ N, Pi ∈ ∆(W). Two assumptions:• [s] is measurable for all strategy profiles s ∈ S• Pi([si]) > 0 for all si ∈ Si
Eric Pacuit 21
ex interim beliefs: Pi,w ∈ ∆(S−i)
I ...given player i’s choice: Pi,w(·) = Pi(· | [si(w)])
I ...given all player i knows: Pi,w(·) = Pi(· | Ki), Ki ⊆ [si(w)]
I ...given all player i fully believes: Pi,w(·) = Pi(· | Bi), Bi ⊆ [si(w)]
Expected utility: Given P ∈ ∆(S−i),
EUi,P(a) =∑
s−i∈S−i
P(s−i)ui(a, s−i)
Eric Pacuit 22
ex interim beliefs: Pi,w ∈ ∆(S−i)
I ...given player i’s choice: Pi,w(·) = Pi(· | [si(w)])
I ...given all player i knows: Pi,w(·) = Pi(· | Ki), Ki ⊆ [si(w)]
I ...given all player i fully believes: Pi,w(·) = Pi(· | Bi), Bi ⊆ [si(w)]
Expected utility: Given P ∈ ∆(S−i),
EUi,P(a) =∑
s−i∈S−i
P(s−i)ui(a, s−i)
Eric Pacuit 22
ex interim beliefs: Pi,w ∈ ∆(S−i)
I ...given player i’s choice: Pi,w(·) = Pi(· | [si(w)])
I ...given all player i knows: Pi,w(·) = Pi(· | Ki), Ki ⊆ [si(w)]
I ...given all player i fully believes: Pi,w(·) = Pi(· | Bi), Bi ⊆ [si(w)]
Expected utility: Given w ∈ W ,
EUi,w(a) =∑
s−i∈S−i
Pi,w([s−i])ui(a, s−i)
Eric Pacuit 22
B
A
U L R
U 2,1 0,0 U
D 0,0 1,2 U
Strategic Game
pA (U)
pB(R)
1
10
23
13
Solution Space
Eric Pacuit 23
B
A
U L R
U 2,1 0,0 U
D 0,0 1,2 U
Strategic Game
pA (U)
pB(R)
1
10
Solution Space
Eric Pacuit 23
B
A
U L R
U 2,1 0,0 U
D 0,0 1,2 U
Strategic Game
pA (U)
pB(R)
1
10
Solution Space
Eric Pacuit 23
B
A
U L R
U 2,1 0,0 U
D 0,0 1,2 U
Strategic Game
pA (U)
pB(R)
1
10
Solution Space
Eric Pacuit 23
B
A
U L R
U 2,1 0,0 U
D 0,0 1,2 U
Strategic Game
pA (U)
pB(R)
1
10
Solution Space
Eric Pacuit 23
B
A
U L R
U 2,1 0,0 U
D 0,0 1,2 U
Strategic Game
pA (U)
pB(R)
1
10
Solution Space
Eric Pacuit 23
B
A
U L R
U 2,1 0,0 U
D 0,0 1,2 U
Strategic Game
pA (U)
pB(R)
1
10
23
23
Solution Space
Eric Pacuit 23
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
U
D
L R
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
I Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
I Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
I Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · PA (L) + 0 · PA (R) ≥ 0 · PA (L) + 2 ·PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
I Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · 34 + 0 · PA (R) ≥ 0 · 3
4 + 2 · PA (R)
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
I Ann’s choice is optimal(given her information)
Bob’s choice is optimal(given her information)
Ann considers it possibleBob is irrational
1 · 34 + 0 · 1
4 ≥ 0 · 34 + 2 · 1
4
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
16
0
312
0
16
112
I Ann’s choice is optimal(given her information)
I Bob’s choice is optimal(given his information)
Ann considers it possibleBob is irrational
2 · 23 + 0 · 1
3 ≥ 0 · 23 + 1 · 1
3
Eric Pacuit 24
An Example
BobA
nn
U L R
U 1,2 0,0 U
D 0,0 2,1 U
16
16
0
0
16
312
0
16
112
I Ann’s choice is optimal(given her information)
I Bob’s choice is optimal(given his information)
I Bob considers it possibleAnn is irrational
0 · 34 + 2 · 1
4 � 1 · 34 + 0 · 1
4
Eric Pacuit 24
For any P ∈ ∆(S−i) and si ∈ Si , EUi,P(si) =∑
s−i∈S−iP(s−i)ui(si , s−i)
For any w ∈ W and si ∈ Si , EUi,w(si) =∑
s−i∈S−iPi,w([s−i])ui(si , s−i)
Rati = {w | EUi,w(si(w)) ≥ EUi,w(si) for all si ∈ Si}
Each P ∈ ∆(W) is associated with PS ∈ ∆(S) as follows: for all s ∈ S,PS(s) = P([s])
Eric Pacuit 25
For any P ∈ ∆(S−i) and si ∈ Si , EUi,P(si) =∑
s−i∈S−iP(s−i)ui(si , s−i)
For any w ∈ W and si ∈ Si , EUi,w(si) =∑
s−i∈S−iPi,w([s−i])ui(si , s−i)
Rati = {w | EUi,w(si(w)) ≥ EUi,w(si) for all si ∈ Si}
Each P ∈ ∆(W) is associated with PS ∈ ∆(S) as follows: for all s ∈ S,PS(s) = P([s])
Eric Pacuit 25
For any P ∈ ∆(S−i) and si ∈ Si , EUi,P(si) =∑
s−i∈S−iP(s−i)ui(si , s−i)
For any w ∈ W and si ∈ Si , EUi,w(si) =∑
s−i∈S−iPi,w([s−i])ui(si , s−i)
Rati = {w | EUi,w(si(w)) ≥ EUi,w(si) for all si ∈ Si}
Each P ∈ ∆(W) is associated with PS ∈ ∆(S) as follows: for all s ∈ S,PS(s) = P([s])
Eric Pacuit 25
For any P ∈ ∆(S−i) and si ∈ Si , EUi,P(si) =∑
s−i∈S−iP(s−i)ui(si , s−i)
For any w ∈ W and si ∈ Si , EUi,w(si) =∑
s−i∈S−iPi,w([s−i])ui(si , s−i)
Rati = {w | EUi,w(si(w)) ≥ EUi,w(si) for all si ∈ Si}
Each P ∈ ∆(W) is associated with PS ∈ ∆(S) as follows: for all s ∈ S,PS(s) = P([s])
Eric Pacuit 25
Mixed Strategy
A mixed strategy is a sequence of probabilities over the players’available actions: σ ∈ Πi∈N∆(Si), Pσ ∈ ∆(S).
Choices are assumed to be independent: Pσ(s) = σ1(s1) · · ·σn(sn)
Eric Pacuit 26
Mixed Strategies
“We are reluctant to believe that our decisions are made at random. Weprefer to be able to point to a reason for each action we take. Outside ofLas Vegas we do not spin roulettes.”
I One can think about a game as an interaction between largepopulations...a mixed strategy is viewed as the distribution of thepure choices in the population.
I Harsanyi’s purification theorem: A player’s mixed strategy is thoughtof as a plan of action which is dependent on private informationwhich is not specified in the model. Although the player’s behaviorappears to be random, it is actually deterministic.
I Mixed strategies are beliefs held by all other players concerning aplayer’s actions.
Eric Pacuit 27
Mixed Strategies
“We are reluctant to believe that our decisions are made at random. Weprefer to be able to point to a reason for each action we take. Outside ofLas Vegas we do not spin roulettes.”
I One can think about a game as an interaction between largepopulations...a mixed strategy is viewed as the distribution of thepure choices in the population.
I Harsanyi’s purification theorem: A player’s mixed strategy is thoughtof as a plan of action which is dependent on private informationwhich is not specified in the model. Although the player’s behaviorappears to be random, it is actually deterministic.
I Mixed strategies are beliefs held by all other players concerning aplayer’s actions.
Eric Pacuit 27
Mixed Strategies
“We are reluctant to believe that our decisions are made at random. Weprefer to be able to point to a reason for each action we take. Outside ofLas Vegas we do not spin roulettes.”
I One can think about a game as an interaction between largepopulations...a mixed strategy is viewed as the distribution of thepure choices in the population.
I Harsanyi’s purification theorem: A player’s mixed strategy is thoughtof as a plan of action which is dependent on private informationwhich is not specified in the model. Although the player’s behaviorappears to be random, it is actually deterministic.
I Mixed strategies are beliefs held by all other players concerning aplayer’s actions.
Eric Pacuit 27
Mixed Strategies
“We are reluctant to believe that our decisions are made at random. Weprefer to be able to point to a reason for each action we take. Outside ofLas Vegas we do not spin roulettes.”
I One can think about a game as an interaction between largepopulations...a mixed strategy is viewed as the distribution of thepure choices in the population.
I Harsanyi’s purification theorem: A player’s mixed strategy is thoughtof as a plan of action which is dependent on private informationwhich is not specified in the model. Although the player’s behaviorappears to be random, it is actually deterministic.
I Mixed strategies are beliefs held by all other players concerning aplayer’s actions.
Eric Pacuit 27
Nash Equilibrium
Let 〈N, {Si}i∈N , {ui}i∈N〉 be a strategic game
For s−i ∈ S−i , let
Bi(s−i) = {si ∈ Si | ui(s−i , si) ≥ ui(s−i , s′i ) ∀ s′i ∈ Si}
Bi is the best-response function.
s∗ ∈ S is a Nash equilibrium iff s∗i ∈ Bi(s∗−i) for all i ∈ N.
Eric Pacuit 28
Nash Equilibrium
Let 〈N, {Si}i∈N , {ui}i∈N〉 be a strategic game
For s−i ∈ S−i , let
Bi(s−i) = {si ∈ Si | ui(s−i , si) ≥ ui(s−i , s′i ) ∀ s′i ∈ Si}
Bi is the best-response function.
s∗ ∈ S is a Nash equilibrium iff s∗i ∈ Bi(s∗−i) for all i ∈ N.
Eric Pacuit 28
Nash Equilibrium: Example
L RU 2,1 0,0D 0,0 1,2
N = {r , c} Ar = {br , sr },Ac = {bc , sc}
Br(bc) = {br } Br(sc) = {sr }
Bc(br) = {bc} Bc(sr) = {sc}
(br , bc) is a Nash Equilibrium (sr , sc) is a Nash Equilibrium
Eric Pacuit 29
Nash Equilibrium: Example
L RU 2,1 0,0D 0,0 1,2
N = {r , c} Sr = {U,D}, Sc = {L ,R}
Br(L) = {U} Br(R) = {D}
Bc(br) = {bc} Bc(sr) = {sc}
(br , bc) is a Nash Equilibrium (sr , sc) is a Nash Equilibrium
Eric Pacuit 29
Nash Equilibrium: Example
L RU 2,1 0,0D 0,0 1,2
N = {r , c} Sr = {U,D}, Sc = {L ,R}
Br(L) = {U} Br(R) = {D}
Bc(U) = {L} Bc(D) = {R}
(br , bc) is a Nash Equilibrium (sr , sc) is a Nash Equilibrium
Eric Pacuit 29
Nash Equilibrium: Example
L RU 2,1 0,0D 0,0 1,2
N = {r , c} Sr = {U,D}, Sc = {L ,R}
Br(L) = {U} Br(R) = {D}
Bc(U) = {L} Bc(D) = {R}
(br , bc) is a Nash Equilibrium (sr , sc) is a Nash Equilibrium
Eric Pacuit 29
Nash Equilibrium: Example
L RU 2,1 0,0D 0,0 1,2
N = {r , c} Sr = {U,D}, Sc = {L ,R}
Br(L) = {U} Br(R) = {D}
Bc(U) = {L} Bc(D) = {R}
(U, L) is a Nash Equilibrium (D,R) is a Nash Equilibrium
Eric Pacuit 29
Battle of the Sexes
Bob
Ann
U B S
B 2, 1 0, 0 U
S 0, 0 1, 2 U
(D,S) and (S,D) are Nash equilibria. If both choose their componentsof these equilibria, we may end up at (D,D).
Eric Pacuit 30
Battle of the Sexes
Bob
Ann
U B M
B 2, 1 0, 0 U
S 0, 0 1, 2 U
(B ,B) (S,S), and ([2/3 : B , 1/3 : S], [1/3 : B , 2/3 : S]) are Nashequilibria.
Eric Pacuit 30
The Guessing Game
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
What number should you guess? 100, 99, . . . , 67, . . . , 2, 1
Eric Pacuit 31
The Guessing Game
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
What number should you guess? 100, 99, . . . , 67, . . . , 2, 1
Eric Pacuit 31
The Guessing Game
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
What number should you guess? 100, 99, . . . , 67, . . . , 2, 1
Eric Pacuit 31
The Guessing Game
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
What number should you guess? ��HH100, 99, . . . , 67, . . . , 2, 1
Eric Pacuit 31
The Guessing Game
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
What number should you guess? ��HH100,��ZZ99, . . . , 67, . . . , 2, 1
Eric Pacuit 31
The Guessing Game
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
What number should you guess? ��HH100,��ZZ99, . . . ,��ZZ67, . . . , 2, 1
Eric Pacuit 31
The Guessing Game
Guess a number between 1 & 100.The closest to 2/3 of the average wins.
What number should you guess? ��HH100,��ZZ99, . . . ,��ZZ67, . . . , �A2, 1
Eric Pacuit 31
Traveler’s Dilemma
2 3 4 · · · 99 1002 (2, 2) (4, 0) (4, 0) · · · (4, 0) (4, 0)
3 (0, 4) (3, 3) (5, 1) · · · (5, 1) (5, 1)
4 (0, 4) (1, 5) (4, 4) · · · (6, 2) (6, 2)
......
......
......
...
99 (0, 4) (1, 5) (2, 6) · · · (99, 99) (101, 97)
100 (0, 4) (1, 5) (2, 6) · · · (97, 101) (100, 100)
(2, 2) is the only Nash equilibrium.
Eric Pacuit 32
Traveler’s Dilemma
2 3 4 · · · 99 1002 (2, 2) (4, 0) (4, 0) · · · (4, 0) (4, 0)
3 (0, 4) (3, 3) (5, 1) · · · (5, 1) (5, 1)
4 (0, 4) (1, 5) (4, 4) · · · (6, 2) (6, 2)
......
......
......
...
99 (0, 4) (1, 5) (2, 6) · · · (99, 99) (101, 97)
100 (0, 4) (1, 5) (2, 6) · · · (97, 101) (100, 100)
(2, 2) is the only Nash equilibrium.
Eric Pacuit 32
Traveler’s Dilemma
2 3 4 · · · 99 1002 (2, 2) (4, 0) (4, 0) · · · (4, 0) (4, 0)
3 (0, 4) (3, 3) (5, 1) · · · (5, 1) (5, 1)
4 (0, 4) (1, 5) (4, 4) · · · (6, 2) (6, 2)
......
......
......
...
99 (0, 4) (1, 5) (2, 6) · · · (99, 99) (101, 97)
100 (0, 4) (1, 5) (2, 6) · · · (97, 101) (100, 100)
The analysis is insensitive to the amount of reward/punishment.
Eric Pacuit 32
Characterizing Nash Equilibria
Theorem (Aumann). σ is a Nash equilibrium of G iff there exists a modelMG = 〈W , {Pi}i∈N , s〉 such that:I for all i ∈ N, Rati = W ;I for all i, j ∈ N, Pi = Pj ; andI for all i ∈ N, PS
i = Pσ.
Eric Pacuit 33
Playing a Nash equilibrium is required by the players rationality andcommon knowledge thereof.
I Players need not be certain of the other players’ beliefsI Strategies that are not an equilibrium may be rationalizableI Sometimes considerations of riskiness trump the Nash equilibrium
Eric Pacuit 34
Bob
Ann
U L C R
T 1, 1 2, 0 -2, 1 U
M 0, 2 1, 1 2, 1 U
B 1, -2 1, 2 1, 1 U
(T , L) is the unique pure-strategy Nash equilibrium
Eric Pacuit 35
Bob
Ann
U L C R
T 1, 1 2, 0 -2, 1 U
M 0, 2 1, 1 2, 1 U
B 1, -2 1, 2 1, 1 U
(T , L) is the unique pure-strategy Nash equilibrium
Eric Pacuit 35
Bob
Ann
U L C R
T 1, 1 2, 0 -2, 1 U
M 0, 2 1, 1 2, 1 U
B 1, -2 1, 2 1, 1 U
(T , L) is the unique pure-strategy Nash equilibrium
Eric Pacuit 35
Bob
Ann
U L C R
T 1, 1 2, 0 -2, 1 U
M 0, 2 1, 1 2, 1 U
B 1, -2 1, 2 1, 1 U
Why not play B and R?
Eric Pacuit 35
Rationalizability
Eric Pacuit 36
Bob
Ann
U L C R
T 3, 2 0, 0 2, 3 U
M 0, 0 1, 1 0, 0 U
B 2, 3 0, 0 3, 2 U
(M,C) is the unique Nash equilibrium
Eric Pacuit 37
Bob
Ann
U L C R
T 3, 2 0, 0 2, 3 U
M 0, 0 1, 1 0, 0 U
B 2, 3 0, 0 3, 2 U
(M,C) is the unique Nash equilibrium
Eric Pacuit 37
Bob
Ann
U L C R
T 3, 2 0, 0 2, 3 U
M 0, 0 1, 1 0, 0 U
B 2, 3 0, 0 3, 2 U
T , L , B and R are rationalizable
Eric Pacuit 37
Bob
Ann
U L C R
T 3, 2 0, 0 2, 3 U
M 0, 0 1, 1 0, 0 U
B 2, 3 0, 0 3, 2 U
T , L , B and R are rationalizable
Eric Pacuit 37
Bob
Ann
U L C R
T 3, 2 0, 0 2, 3 U
M 0, 0 1, 1 0, 0 U
B 2, 3 0, 0 3, 2 U
Ann plays B because she thought Bob will play R
Eric Pacuit 37
Bob
Ann
U L C R
T 3, 2 0, 0 2, 3 U
M 0, 0 1, 1 0, 0 U
B 2, 3 0, 0 3, 2 U
Bob plays L because she thought Ann will play B
Eric Pacuit 37
Bob
Ann
U L C R
T 3, 2 0, 0 2, 3 U
M 0, 0 1, 1 0, 0 U
B 2, 3 0, 0 3, 2 U
Bob was correct, but Ann was wrong
Eric Pacuit 37
Bob
Ann
U L C R X
T 3, 2 0, 0 2, 3 0, -5 U
M 0, 0 1, 1 0, 0 200,-5 U
B 2, 3 0, 0 3, 2 1,-3 U
Not every strategy is rationalizable: Ann can’t play M because shethinks Bob will play X
Eric Pacuit 37
“Analysis of strategic economic situations requires us, implicitly orexplicitly, to maintain as plausible certain psychological hypotheses. Thehypothesis that real economic agents universally recognize the salienceof Nash equilibria may well be less accurate than, for example, thehypothesis that agents attempt to “out-smart” or “second-guess” eachother, believing that their opponents do likewise.” (pg. 1010)
B. D. Bernheim. Rationalizable Strategic Behavior. Econometrica, 52:4, pgs. 1007 - 1028,1984.
Eric Pacuit 38
“The rules of a game and its numerical data are seldom sufficient forlogical deduction alone to single out a unique choice of strategy for eachplayer. To do so one requires either richer information (such asinstitutional detail or perhaps historical precedent for a certain type ofbehavior) or bolder assumptions about how players choose strategies.Putting further restrictions on strategic choice is a complex andtreacherous task. But one’s intuition frequently points to patterns ofbehavior that cannot be isolated on the grounds of consistency alone.”asdlfsadf (pg. 1035)
D. G. Pearce. Rationalizable Strategic Behavior. Econometrica, 52, 4, pgs. 1029 - 1050,1984.
Eric Pacuit 39
Rationalizability
A best reply set (BRS) is a sequence (B1,B2, . . . ,Bn) ⊆ S = Πi∈NSi
such that for all i ∈ N, and all si ∈ Bi , there exists µ−i ∈ ∆(B−i) such thatsi is a best response to µ−i : I.e.,
bi = arg maxsi∈Si
EUi,µ−i (si)
.
Eric Pacuit 40
2
1
b1 b2 b3 b4
a1 0, 7 2, 5 7, 0 0, 1a2 5, 2 3, 3 5, 2 0, 1a3 7, 0 2, 5 0, 7 0, 1a4 0, 0 0, -2 0, 0 10, -1
I (a2, b2) is the unique Nash equilibria, hence ({a2}, {b2}) is a BRSI ({a1, a3}, {b1, b3}) is a BRSI ({a1, a2, a3}, {b1, b2, b3}) is a full BRS
Eric Pacuit 41
2
1
b1 b2 b3 b4
a1 0, 7 2, 5 7, 0 0, 1a2 5, 2 3, 3 5, 2 0, 1a3 7, 0 2, 5 0, 7 0, 1a4 0, 0 0, -2 0, 0 10, -1
I (a2, b2) is the unique Nash equilibria, hence ({a2}, {b2}) is a BRS
I ({a1, a3}, {b1, b3}) is a BRSI ({a1, a2, a3}, {b1, b2, b3}) is a full BRS
Eric Pacuit 41
2
1
b1 b2 b3 b4
a1 0, 7 2, 5 7, 0 0, 1a2 5, 2 3, 3 5, 2 0, 1a3 7, 0 2, 5 0, 7 0, 1a4 0, 0 0, -2 0, 0 10, -1
I (a2, b2) is the unique Nash equilibria, hence ({a2}, {b2}) is a BRSI ({a1, a3}, {b1, b3}) is a BRS
I ({a1, a2, a3}, {b1, b2, b3}) is a full BRS
Eric Pacuit 41
2
1
b1 b2 b3 b4
a1 0, 7 2, 5 7, 0 0, 1a2 5, 2 3, 3 5, 2 0, 1a3 7, 0 2, 5 0, 7 0, 1a4 0, 0 0, -2 0, 0 10, -1
I (a2, b2) is the unique Nash equilibria, hence ({a2}, {b2}) is a BRSI ({a1, a3}, {b1, b3}) is a BRSI ({a1, a2, a3}, {b1, b2, b3}) is a full BRS
Eric Pacuit 41
Theorem (Bernheim; Pearce; Brandenburger and Dekel; . . . ).(B1,B2, . . . ,Bn) is a BRS for G iff there exists a model such that theprojection of the the event where there is common belief that all playersare rational is B1 × · · · × Bn.
Eric Pacuit 42
BobA
nnU L R
U 2,2 4,1 U
D 1,4 3,3 U
Game 1
Bob
Ann
U L R
U 2,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: D strictly dominates U and R strictly dominates L .
Game 2: U strictly dominates D, and after removing D, L strictlydominates R.
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 43
BobA
nnU L R
U 2,2 4,1 U
D 1,4 3,3 U
Game 1
Bob
Ann
U L R
U 2,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U strictly dominates D and L strictly dominates R.
Game 2: U strictly dominates D, and after removing D, L strictlydominates R.
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 43
BobA
nnU L R
U 2,2 4,1 U
D 1,4 3,3 U
Game 1
Bob
Ann
U L R
U 2,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U strictly dominates D and L strictly dominates R.
Game 2: U strictly dominates D, and after removing D, L strictlydominates R.
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 43
BobA
nnU L R
U 2,2 4,1 U
D 1,4 3,3 U
Game 1
Bob
Ann
U L R
U 2,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U strictly dominates D and L strictly dominates R.
Game 2: U strictly dominates D, and after removing D, L strictlydominates R.
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 43
BobA
nnU L R
U 2,2 4,1 U
D 1,4 3,3 U
Game 1
Bob
Ann
U L R
U 2,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U strictly dominates D and L strictly dominates R.
Game 2: U strictly dominates D, and after removing D, L strictlydominates R.
Theorem. In all models where the players are rational and there iscommon belief of rationality, the players choose strategies that surviveiterative removal of strictly dominated strategies (and, conversely...).
Eric Pacuit 43
Bob
Ann
U L R
U 3,3 1,1 U
D 2,2 2,2 U
Is R rationalizable?
There is no full support probability such that R is a best responseShould Ann assign probability 0 to R or probability > 0 to R?
Eric Pacuit 44
Bob
Ann
U L R
U 3,3 1,1 U
D 2,2 2,2 U
Is R rationalizable?There is no full support probability such that R is a best response
Should Ann assign probability 0 to R or probability > 0 to R?
Eric Pacuit 44
Bob
Ann
U L R
U 3,3 1,1 U
D 2,2 2,2 U
Is R rationalizable?There is no full support probability such that R is a best responseShould Ann assign probability 0 to R or probability > 0 to R?
Eric Pacuit 44
Strategic Reasoning and Admissibility
“The argument for deletion of a weakly dominated strategy for player i isthat he contemplates the possibility that every strategy combination ofhis rivals occurs with positive probability. However, this hypothesisclashes with the logic of iterated deletion, which assumes, precisely, thateliminated strategies are not expected to occur.”
Mas-Colell, Whinston and Green. Introduction to Microeconomics. 1995.
Eric Pacuit 45
A Puzzle
R. Cubitt and R. Sugden. Rationally Justifiable Play and the Theory of Non-cooperativegames. Economic Journal, 104, pgs. 798 - 803, 1994.
R. Cubitt and R. Sugden. Common reasoning in games: A Lewisian analysis of commonknowledge of rationality. Manuscript, 2011.
Eric Pacuit 46
BobA
nnU L R
U 1,1 0,0 U
D 0,0 0,0 U
Game 1
Bob
Ann
U L R
U 1,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U weakly dominates D and L weakly dominates R.
Game 2: U strictly dominates D, and after removing D, L strictlydominates R.
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 47
BobA
nnU L R
U 1,1 0,0 U
D 0,0 0,0 U
Game 1
Bob
Ann
U L R
U 1,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U weakly dominates D and L weakly dominates R.
Game 2: U weakly dominates D, and after removing D, L strictlydominates R.
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 47
BobA
nnU L R
U 1,1 0,0 U
D 0,0 0,0 U
Game 1
Bob
Ann
U L R
U 1,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U weakly dominates D and L weakly dominates R.
Game 2: U weakly dominates D, and after removing D, L strictlydominates R.
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 47
BobA
nnU L R
U 1,1 0,0 U
D 0,0 0,0 U
Game 1
Bob
Ann
U L R
U 1,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U weakly dominates D and L weakly dominates R.
Game 2: U weakly dominates D, and after removing D, L strictlydominates R.
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 47
BobA
nnU L R
U 1,1 0,0 U
D 0,0 0,0 U
Game 1
Bob
Ann
U L R
U 1,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U weakly dominates D and L weakly dominates R.
Game 2: But, now what is the reason for not playing D?
Theorem. The projection of any event where the players are rational andthere is common belief of rationality are strategies that survive iterativeremoval of strictly dominated strategies (and, conversely...).
Eric Pacuit 47
BobA
nnU L R
U 1,1 0,0 U
D 0,0 0,0 U
Game 1
Bob
Ann
U L R
U 1,1 1,0 U
D 1,0 0,1 U
Game 2
Game 1: U weakly dominates D and L weakly dominates R.
Game 2: But, now what is the reason for not playing D?
Theorem (Samuelson). There is no model of Game 2 satisfying commonknowledge of rationality (where rationality incorporates weakdominance). adfa sfas df adsf asd fa
Eric Pacuit 47
Common Knowledge of Admissibility
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
T , L T ,R T , {L ,R}
B , L B ,R B , {L ,R}
{T ,B}, L {T ,B},R {T ,B}, {L ,R}
There is no model of this game with common knowledge of admissibility.
Eric Pacuit 48
Common Knowledge of Admissibility
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
T , L T ,R T , {L ,R}
B , L B ,R B , {L ,R}
{T ,B}, L {T ,B},R {T ,B}, {L ,R}
The ”full” model of the game: B is not admissible given Ann’s information
Eric Pacuit 48
Common Knowledge of Admissibility
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
T , L T ,R T , {L ,R}
B , L B ,R B , {L ,R}
{T ,B}, L {T ,B},R {T ,B}, {L ,R}
The ”full” model of the game: B is not admissible given Ann’s information
Eric Pacuit 48
Common Knowledge of Admissibility
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
T , L T ,R T , {L ,R}
B , L B ,R B , {L ,R}
{T ,B}, L {T ,B},R {T ,B}, {L ,R}
What is wrong with this model? asdf ad fa sdf a fsd asdf adsf adfs
Eric Pacuit 48
Common Knowledge of Admissibility
Bob
Ann
T L R
T 1,1 1,0 U
B 1,0 0,1 U
T , L T ,R T , {L ,R}
B , L B ,R B , {L ,R}
{T ,B}, L {T ,B},R {T ,B}, {L ,R}
Privacy of Tie-Breaking/No Extraneous Beliefs: If a strategy isrational for an opponent, then it cannot be “ruled out”.
Eric Pacuit 48
The Epistemic Program in Game Theory
Game G
Strategy Space
Ann’s States Bob’s States
G: available actions, payoffs, structure of the decision problem
Eric Pacuit 49
The Epistemic Program in Game Theory
Game G
Strategy Space
Ann’s States Bob’s States
solution concepts are systematic descriptions of what players do
Eric Pacuit 49
The Epistemic Program in Game Theory
Game G
Strategy Space
Ann’s States Bob’s States
The game model includes information states of the players
Eric Pacuit 49
The Epistemic Program in Game Theory
Game G
Strategy Space
Ann’s States Bob’s States
Restrict to information states satisfying some rationality condition
Eric Pacuit 49
The Epistemic Program in Game Theory
Game G
Strategy Space
Ann’s States Bob’s States
Project onto the strategy space
Eric Pacuit 49
Why go beyond the basic Bayesian model?
I Counterfactual beliefs/choices are important for assessing therationality of a strategy.
I Static models of dynamic games: A game model represents howa player will and would change her beliefs if her opponents take thegame in various directions.
I A conditional choice (do a if E) is rational iff doing a would berational if the player were to learn E.
I Strategy choices should be robust against mistaken beliefs(weak dominance).
Eric Pacuit 50
Why go beyond the basic Bayesian model?
I Counterfactual beliefs/choices are important for assessing therationality of a strategy.
I Static models of dynamic games: A game model represents howa player will and would change her beliefs if her opponents take thegame in various directions.
I A conditional choice (do a if E) is rational iff doing a would berational if the player were to learn E.
I Strategy choices should be robust against mistaken beliefs(weak dominance).
Eric Pacuit 50
Why go beyond the basic Bayesian model?
I Counterfactual beliefs/choices are important for assessing therationality of a strategy.
“Even if I think you know what I am going to do, I can consider how Ithink you would react if I did something that you and I both know Iwill not do, and my answers to such counterfactual questions will berelevant to assessing the rationality of what I am going to do.” Pasdf(pg. 31)
R. Stalnaker. Belief revision in games: forward and backward induction.Mathematical Social Sciences, 36, pgs. 31 - 56, 1998.
I Static models of dynamic games: A game model represents howa player will and would change her beliefs if her opponents take thegame in various directions.
I A conditional choice (do a if E) is rational iff doing a would berational if the player were to learn E.
I Strategy choices should be robust against mistaken beliefs(weak dominance).
Eric Pacuit 50
Why go beyond the basic Bayesian model?
I Counterfactual beliefs/choices are important for assessing therationality of a strategy.
I Static models of dynamic games: A game model represents howa player will and would change her beliefs if her opponents take thegame in various directions.
I A conditional choice (do a if E) is rational iff doing a would berational if the player were to learn E.
I Strategy choices should be robust against mistaken beliefs(weak dominance).
Eric Pacuit 50
Why go beyond the basic Bayesian model?
I Counterfactual beliefs/choices are important for assessing therationality of a strategy.
I Static models of dynamic games: A game model represents howa player will and would change her beliefs if her opponents take thegame in various directions.
I A conditional choice (do a if E) is rational iff doing a would berational if the player were to learn E.
I Strategy choices should be robust against mistaken beliefs(weak dominance).
Eric Pacuit 50
Why go beyond the basic Bayesian model?
I Counterfactual beliefs/choices are important for assessing therationality of a strategy.
I Static models of dynamic games: A game model represents howa player will and would change her beliefs if her opponents take thegame in various directions.
I A conditional choice (do a if E) is rational iff doing a would berational if the player were to learn E.
I Strategy choices should be robust against mistaken beliefs(weak dominance).
Eric Pacuit 50
Why go beyond the basic Bayesian model?
I Counterfactual beliefs/choices are important for assessing therationality of a strategy.
I Static models of dynamic games: A game model represents howa player will and would change her beliefs if her opponents take thegame in various directions.
I A conditional choice (do a if E) is rational iff doing a would berational if the player were to learn E.
I Strategy choices should be robust against mistaken beliefs(weak dominance).
Eric Pacuit 50
In a game modelMG = 〈W , {Pi}i∈N , s〉, different states representdifferent beliefs only when the agent is doing something different.
Pi,w(E) = Pi(E | [si(w)])
To represent different explanations (i.e., beliefs) for the same strategychoice, we would need a set of models {MG
1 ,MG2 , . . .}.
Eric Pacuit 51
In a game modelMG = 〈W , {Pi}i∈N , s〉, different states representdifferent beliefs only when the agent is doing something different.
Pi,w(E) = Pi(E | Bi,w), Bi,w ⊆ [si(w)]
To represent different explanations (i.e., beliefs) for the same strategychoice, we would need a set of models {MG
1 ,MG2 , . . .}.
Eric Pacuit 51
In a game modelMG = 〈W , {Pi}i∈N , s〉, different states representdifferent beliefs only when the agent is doing something different.
Pi,w(E) = Pi(E | Bi,w), Bi,w ⊆ [si(w)]
Two way to change beliefs: Pi(· | E ∩ Bi,w) and Pi(· | B′i,w) (conditioningon 0 events).
Eric Pacuit 51
Game Models
Richer models of a game: lexicographic probabilities, conditionalprobability systems, non-standard probabilities, plausibility models, . . .(type spaces)
“The aim in giving the general definition of a model is not to propose anoriginal explanatory hypothesis, or any explanatory hypothesis, for thebehavior of players in games, but only to provide a descriptive frameworkfor the representation of considerations that are relevant to suchexplanations, a framework that is as general and as neutral as we canmake it.” (pg. 35)
R. Stalnaker. Knowledge, Belief and Counterfactual Reasoning in Games. Economicsand Philosophy, 12(1), pgs. 133 - 163, 1996.
Eric Pacuit 52
Game Models
Richer models of a game: lexicographic probabilities, conditionalprobability systems, non-standard probabilities, plausibility models, . . .(type spaces)
“The aim in giving the general definition of a model is not to propose anoriginal explanatory hypothesis, or any explanatory hypothesis, for thebehavior of players in games, but only to provide a descriptive frameworkfor the representation of considerations that are relevant to suchexplanations, a framework that is as general and as neutral as we canmake it.” (pg. 35)
R. Stalnaker. Knowledge, Belief and Counterfactual Reasoning in Games. Economicsand Philosophy, 12(1), pgs. 133 - 163, 1996.
Eric Pacuit 52
Richer models of games
1. A partition ≈i representing the different “types” of player i: w ≈i vmeans that w and v are subjectively indistinguishable to player i(i’s beliefs, knowledge, and conditional beliefs are the same in bothstates).
2. A pseudo-partition Ri (serial, transitive and Euclidean relation)representing a player i’s working hypotheses (full beliefs?, seriouspossibilities?,. . . ).
3. Player i’s belief revision policy described in terms of i’s conditionalbeliefs.
This can all be represented by a single relation �i ⊆ W ×W
Eric Pacuit 53
Richer models of games
1. A partition ≈i representing the different “types” of player i: w ≈i vmeans that w and v are subjectively indistinguishable to player i(i’s beliefs, knowledge, and conditional beliefs are the same in bothstates).
2. A pseudo-partition Ri (serial, transitive and Euclidean relation)representing a player i’s working hypotheses (full beliefs?, seriouspossibilities?,. . . ).
3. Player i’s belief revision policy described in terms of i’s conditionalbeliefs.
This can all be represented by a single relation �i ⊆ W ×W
Eric Pacuit 53
Richer models of games
MG = 〈W , {�i ,Pi}i∈N , s〉, where W , Pi and s are as before and �i is areflexive, transitive and locally-connected relation.
1. w ≈i v iff w �i v or v �i w. Let [w]≈i = {v | w ≈i v}
2. w Ri v iff v ∈ Max�i ([w]≈i )
3. Bi,w(F) = Max�i (F ∩ [w]≈i )
Pi,w(E | F) = Pi(E | Bi,w(F))
Eric Pacuit 53
Belief Revision in Games
Exactly how a player revises her beliefs during the game depends, inpart, on how she interprets the observed moves of her opponents.
Eric Pacuit 54
Interpreting Moves
How should Bob respond given evidence that Ann’s moves seemirrational?
1. Bob’s beliefs about Ann’s perception of the game are incorrect.
2. Bob’s assumption about Ann’s decision procedure is incorrect.
3. Bob’s belief about Ann’s assumptions about him is incorrect.
4. Ann’s moves are an attempt to influence Bob’s behavior in the game.
5. Ann simply failed to successfully implement her adopted strategy,i.e., Ann made a “trembling hand mistake”.
Eric Pacuit 55
Interpreting Moves
How should Bob respond given evidence that Ann’s moves seemirrational?
1. Bob’s beliefs about Ann’s perception of the game are incorrect.
2. Bob’s assumption about Ann’s decision procedure is incorrect.
3. Bob’s belief about Ann’s assumptions about him is incorrect.
4. Ann’s moves are an attempt to influence Bob’s behavior in the game.
5. Ann simply failed to successfully implement her adopted strategy,i.e., Ann made a “trembling hand mistake”.
Eric Pacuit 55
Interpreting Moves
How should Bob respond given evidence that Ann’s moves seemirrational?
1. Bob’s beliefs about Ann’s perception of the game are incorrect.
2. Bob’s assumption about Ann’s decision procedure is incorrect.
3. Bob’s belief about Ann’s assumptions about him is incorrect.
4. Ann’s moves are an attempt to influence Bob’s behavior in the game.
5. Ann simply failed to successfully implement her adopted strategy,i.e., Ann made a “trembling hand mistake”.
Eric Pacuit 55
Taking Stock
I Game models can be used to characterize different solutionconcepts (e.g., iterated strict dominance, iterated weak dominance,Nash equilibrium, correlated equilibrium, backward induction,extensive-form rationalizability,...)
I Focusing on dynamic games with simultaneous moves:
• strategy choice is not an instantaneous commitment, but, rather, arepresentation of what the players will and would do in the course ofplaying the game
• Epistemic models describe how the players will and would revise herbeliefs during a play of the game
Eric Pacuit 56
Taking Stock
I Game models can be used to characterize different solutionconcepts (e.g., iterated strict dominance, iterated weak dominance,Nash equilibrium, correlated equilibrium, backward induction,extensive-form rationalizability,...)
I Focusing on dynamic games with simultaneous moves:
• strategy choice is not an instantaneous commitment, but, rather, arepresentation of what the players will and would do in the course ofplaying the game
• Epistemic models describe how the players will and would revise herbeliefs during a play of the game
Eric Pacuit 56
Backward and Forward Induction
There are many epistemic characterizations (Aumann, Stalnaker,Battigalli & Siniscalchi, Friedenberg & Siniscalchi, Perea, Baltag &Smets, Bonanno, van Benthem,...)
I How should we compare the two “styles of reasoning” aboutgames? (Heifetz & Perea, Reny, Battigalli & Siniscalchi)
I How do (should) players choose between the two different styles ofreasoning about games? (Perea, EP & Knoks)
Aumann & Dreze: “When all is said and done, how should we playand what should we expect”.
Eric Pacuit 57
Backward and Forward Induction
There are many epistemic characterizations (Aumann, Stalnaker,Battigalli & Siniscalchi, Friedenberg & Siniscalchi, Perea, Baltag &Smets, Bonanno, van Benthem,...)
I How should we compare the two “styles of reasoning” aboutgames? (Heifetz & Perea, Reny, Battigalli & Siniscalchi)
I How do (should) players choose between the two different styles ofreasoning about games? (Perea, EP & Knoks)
Aumann & Dreze: “When all is said and done, how should we playand what should we expect”.
Eric Pacuit 57
BI Puzzle?
A B A
(2,1) (1,6) (7,5)
(6,6)R1 r R2
D1 d D2
I know Ann is rational,but what should I do ifshe’s not...
Eric Pacuit 58
BI Puzzle?
A B A
(2,1) (1,6) (7,5)
(6,6)R1 r R2
D1 d D2
I know Ann is rational,but what should I do ifshe’s not...
Eric Pacuit 58
BI Puzzle?
A B A
(2,1) (1,6) (7,5)
(6,6)R1 r R2
D1 d D2
I know Ann is rational,but what should I do ifshe’s not...
Eric Pacuit 58
A B A 3, 3
2, 2 1, 1 0, 0
L1
T1 t
l
T2
L2
Eric Pacuit 59
(3, 3)A A A 3
2 1 0
L1
T1 t
l
T2
L2
Eric Pacuit 59
(3, 3)A1 A2 A3 3
2 1 0
L1
T1 t
l
T2
L2
Eric Pacuit 59
(3, 3)A B C 3
2 1 0
L1
T1 t
l
T2
L2
Eric Pacuit 59
(3, 3)A B A 3
2 1 0
L1
T1 t
l
T2
L2
Eric Pacuit 59
Bob
Ann
U t l
T 2,2 2,2 U
LT 1,1 0,0 U
LL 1,1 3,3 U
A B A 3, 3
2, 2 1, 1 0, 0
L
T t
l
T
L
Eric Pacuit 59
Materially Rational: every choice actually made is optimal (i.e.,maximizes subjective expected utility).
Substantively Rational: the player is materially rational and, in addition,for each possible choice, the player would have chosen rationally if shehad had the opportunity to choose.
E.g., Taking keys away from someone who is drunk.
Eric Pacuit 60
Materially Rational: every choice actually made is optimal (i.e.,maximizes subjective expected utility).
Substantively Rational: the player is materially rational and, in addition,for each possible choice, the player would have chosen rationally if shehad had the opportunity to choose.
E.g., Taking keys away from someone who is drunk.
Eric Pacuit 60
BobA
nnU t l
T 2,2 2,2 U
LT 1,1 0,0 U
LL 1,1 3,3 U
A B A 3, 3
2, 2 1, 1 0, 0
L
T t
l
T
L
I Bob’s belief in a causal counterfactual: Ann would choose L on hersecond move if she had a chance to move.
I But we need to ask what would Bob believe about Ann if he learned thathe was wrong about her first choice. This is a question about Bob’sbelief revision policy.
Eric Pacuit 61
Informal characterizations of BI
I Future choices are epistemically independent of any observedbehavior
I Any “off-equilibrium” choice is interpreted simply as a mistake(which will not be repeated)
I At each choice point in a game, the players only reason about futurepaths
Eric Pacuit 62
A
2, ∗B
l ru 4, ∗ 0, ∗
InOut
Eric Pacuit 63
A
2, 1B
l ru 4, 1 0, 0
InOut
Eric Pacuit 63
A
2, 1
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
InOut
Eric Pacuit 63
A
2, 1
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
InOut
Eric Pacuit 63
B
A
l rOut 2, 1 2, 1
u 4, 1 0, 0d 0, 0 1, 4
Eric Pacuit 63
B
A
l rOut 2, 1 2, 1
u 4, 1 0, 0d 0, 0 1, 4
Eric Pacuit 63
B
A
l rOut 2, 1 2, 1
u 4, 1 0, 0d 0, 0 1, 4
Eric Pacuit 63
B
A
l rOut 2, 1 2, 1
u 4, 1 0, 0d 0, 0 1, 4
Eric Pacuit 63
B
A
l rOut 2, 1 2, 1u 4, 1 0, 0d 0, 0 1, 4
Eric Pacuit 63
Ann
Bob
Ann
l ru 2, 1 -2, 0d -2, 0 -1, 4
Bob
Ann
l ru 4, 1 0, 0d 0, 0 1, 4
NB
Eric Pacuit 63
Bob
Ann
ll lr rl rrBu 2, 1 2, 1 -2, 0 -2, 0Bd -2, 0 -2, 0 -1, 4 -1, 4Nu 4, 1 0, 0 4, 1 0, 0Nd 0, 0 1, 4 0, 0 1, 4
Eric Pacuit 63
Bob
Ann
ll lr rl rrBu 2, 1 2, 1 -2, 0 -2, 0Bd -2, 0 -2, 0 -1, 4 -1, 4Nu 4, 1 0, 0 4, 1 0, 0Nd 0, 0 1, 4 0, 0 1, 4
Eric Pacuit 63
Bob
Ann
ll lr rl rrBu 2, 1 2, 1 -2, 0 -2, 0Bd -2, 0 -2, 0 -1, 4 -1, 4Nu 4, 1 0, 0 4, 1 0, 0Nd 0, 0 1, 4 0, 0 1, 4
Eric Pacuit 63
Bob
Ann
ll lr rl rrBu 2, 1 2, 1 -2, 0 -2, 0Bd -2, 0 -2, 0 -1, 4 -1, 4Nu 4, 1 0, 0 4, 1 0, 0Nd 0, 0 1, 4 0, 0 1, 4
Eric Pacuit 63
Bob
Ann
ll lr rl rrBu 2, 1 2, 1 -2, 0 -2, 0Bd -2, 0 -2, 0 -1, 4 -1, 4Nu 4, 1 0, 0 4, 1 0, 0Nd 0, 0 1, 4 0, 0 1, 4
Eric Pacuit 63
Bob
Ann
ll lr rl rrBu 2, 1 2, 1 -2, 0 -2, 0Bd -2, 0 -2, 0 -1, 4 -1, 4Nu 4, 1 0, 0 4, 1 0, 0Nd 0, 0 1, 4 0, 0 1, 4
Eric Pacuit 63
Bob
Ann
ll lr rl rrBu 2, 1 2, 1 -2, 0 -2, 0Bd -2, 0 -2, 0 -1, 4 -1, 4Nu 4, 1 0, 0 4, 1 0, 0Nd 0, 0 1, 4 0, 0 1, 4
Eric Pacuit 63
Bob
Ann
ll lr rl rrBu 2, 1 2, 1 -2, 0 -2, 0Bd -2, 0 -2, 0 -1, 4 -1, 4Nu 4, 1 0, 0 4, 1 0, 0Nd 0, 0 1, 4 0, 0 1, 4
Eric Pacuit 63
Ann
Bob
Ann
l ru 2, 1 -2, 0d -2, 0 -1, 4
Bob
Ann
l ru 4, 1 0, 0d 0, 0 1, 4
NB
Eric Pacuit 63
What is forward induction reasoning?
Forward Induction Principle: a player should use all information sheacquired about her opponents’ past behavior in order to improve herprediction of their future simultaneous and past (unobserved) behavior,relying on the assumption that they are rational.
P. Battigalli. On Rationalizability in Extensive Games. Journal of Economic Theory, 74,pgs. 40 - 61, 1997.
Eric Pacuit 64
Four key issues
I Should the analysis take place on the tree or the matrix? (plans vs.strategies)
I The players’ conditional beliefs must be rich enough to employ theforward induction principle.
I Do the players robustly believe the forward induction principle?I Can players become more/less confident in the forward induction
principle?
Eric Pacuit 65
“...in general, a player’s beliefs about what another player will do arebased on an inference from two other kinds of beliefs: beliefs about thepassive beliefs of that player, and beliefs about her rationality.
If one’sprediction based on these beliefs is defeated, one must choose whetherto revise one’s belief about the other players’s beliefs or one’s belief thatshe is rational...But the assumption that the rationalization principle iscommon belief is itself an assumption about the passive beliefs of otherplayers, and so it is itself something that (according to the principle)might have to be given up in the face of surprising behavioralinformation. So the rationalization principle undermines its own stability.”(pg. 51, Stalnaker)
Eric Pacuit 66
“...in general, a player’s beliefs about what another player will do arebased on an inference from two other kinds of beliefs: beliefs about thepassive beliefs of that player, and beliefs about her rationality. If one’sprediction based on these beliefs is defeated, one must choose whetherto revise one’s belief about the other players’s beliefs or one’s belief thatshe is rational...
But the assumption that the rationalization principle iscommon belief is itself an assumption about the passive beliefs of otherplayers, and so it is itself something that (according to the principle)might have to be given up in the face of surprising behavioralinformation. So the rationalization principle undermines its own stability.”(pg. 51, Stalnaker)
Eric Pacuit 66
“...in general, a player’s beliefs about what another player will do arebased on an inference from two other kinds of beliefs: beliefs about thepassive beliefs of that player, and beliefs about her rationality. If one’sprediction based on these beliefs is defeated, one must choose whetherto revise one’s belief about the other players’s beliefs or one’s belief thatshe is rational...But the assumption that the rationalization principle iscommon belief is itself an assumption about the passive beliefs of otherplayers, and so it is itself something that (according to the principle)might have to be given up in the face of surprising behavioralinformation. So the rationalization principle undermines its own stability.”(pg. 51, Stalnaker)
Eric Pacuit 66
A
2, 1
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
InOut
Eric Pacuit 67
A
2, 1
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
InOut
Eric Pacuit 67
A
Bob
Ann
l ru 2, 1 -2, 0d -2, 0 -1, 4
Bob
Ann
l ru 4, 1 0, 0d 0, 0 1, 4
NB
Eric Pacuit 67
A
Bob
Ann
l ru 2, 1 -2, 0d -2, 0 -1, 4
Bob
Ann
l ru 4, 1 0, 0d 0, 0 1, 4
NB
Eric Pacuit 67
“...Only if one assumes a specific infinite hierarchy of belief revisionpriorities can one be sure that unlimited iteration of forward inductionreasoning will work....But it seems to me that such detailed assumptionsabout belief revision policy....have no intuitive plausibility.”assdf (Stalnaker, pg. 53)
Eric Pacuit 68
Algorithm and a “Theorem”
Algorithm: Eliminate weakly dominated strategies for just two rounds,and then eliminate strictly dominated strategies iteratively.
“Theorem”: It can be proved that all and only strategies that survive thisprocess are realizable in sufficiently rich models in which it is commonbelief that all players are rational, and that all revise their beliefs inconformity with the rationalization principle.
Joint work with Aleks Knoks: “Theorem” ↪→ Theorem
Eric Pacuit 69
Algorithm and a “Theorem”
Algorithm: Eliminate weakly dominated strategies for just two rounds,and then eliminate strictly dominated strategies iteratively.
“Theorem”: It can be proved that all and only strategies that survive thisprocess are realizable in sufficiently rich models in which it is commonbelief that all players are rational, and that all revise their beliefs inconformity with the rationalization principle.
Joint work with Aleks Knoks: “Theorem” ↪→ Theorem
Eric Pacuit 69
Algorithm and a “Theorem”
Algorithm: Eliminate weakly dominated strategies for just two rounds,and then eliminate strictly dominated strategies iteratively.
“Theorem”: It can be proved that all and only strategies that survive thisprocess are realizable in sufficiently rich models in which it is commonbelief that all players are rational, and that all revise their beliefs inconformity with the rationalization principle.
Joint work with Aleks Knoks: “Theorem” ↪→ Theorem
Eric Pacuit 69
Algorithm and a “Theorem”
Algorithm: Eliminate weakly dominated strategies for just two rounds,and then eliminate strictly dominated strategies iteratively.
“Theorem”: It can be proved that all and only strategies that survive thisprocess are realizable in sufficiently rich models in which it is commonbelief that all players are rational, and that all revise their beliefs inconformity with the rationalization principle.
Joint work with Aleks Knoks: “Theorem” ↪→ Theorem
Eric Pacuit 69
Algorithm and a “Theorem”
Algorithm: Eliminate weakly dominated strategies for just two rounds,and then eliminate strictly dominated strategies iteratively.
“Theorem”: It can be proved that all and only strategies that survive thisprocess are realizable in sufficiently rich models in which it is commonbelief that all players are rational, and that all revise their beliefs inconformity with the rationalization principle.
Joint work with Aleks Knoks: “Theorem” ↪→ Theorem
Eric Pacuit 69
Backward versus Forward Induction
A
3, 0
Bob
Ann
l c ru 2, 2 2, 1 0, 0d 1, 1 1, 2 4, 0
A. Perea. Backward Induction versus Forward Induction Reasoning. Games, 1, pgs. 168- 188, 2010.
Eric Pacuit 70
Backward versus Forward Induction
A
3, 0
Bob
Ann
l c ru 2, 2 2, 1 0, 0d 1, 1 1, 2 4, 0
A. Perea. Backward Induction versus Forward Induction Reasoning. Games, 1, pgs. 168- 188, 2010.
Eric Pacuit 70
Backward versus Forward Induction
A
3, 0
Bob
Ann
l c ru 2, 2 2, 1 0, 0d 1, 1 1, 2 4, 0
A. Perea. Backward Induction versus Forward Induction Reasoning. Games, 1, pgs. 168- 188, 2010.
Eric Pacuit 70
Backward versus Forward Induction
A
3, 0
Bob
Ann
l c ru 2, 2 2, 1 0, 0d 1, 1 1, 2 4, 0
A. Perea. Backward Induction versus Forward Induction Reasoning. Games, 1, pgs. 168- 188, 2010.
Eric Pacuit 70
Backward versus Forward Induction
A
3, 0
Bob
Ann
l c ru 2, 2 2, 1 0, 0d 1, 1 1, 2 4, 0
A. Perea. Backward Induction versus Forward Induction Reasoning. Games, 1, pgs. 168- 188, 2010.
Eric Pacuit 70
Backward versus Forward Induction
A
3, 0
Bob
Ann
l c ru 2, 2 2, 1 0, 0d 1, 1 1, 2 4, 0
A. Perea. Backward Induction versus Forward Induction Reasoning. Games, 1, pgs. 168- 188, 2010.
Eric Pacuit 70
Backward versus Forward Induction
A
3, 0
Bob
Ann
l c ru 2, 2 2, 1 0, 0d 1, 1 1, 2 4, 0
A. Perea. Backward Induction versus Forward Induction Reasoning. Games, 1, pgs. 168- 188, 2010.
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Rationalization versus Mistakes
A2, 2
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
In
Out
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Rationalization versus Mistakes
A3, 3
A3, 3
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
In2
Out2
In1
Out1
Eric Pacuit 71
Rationalization versus Mistakes
A3, 3
A2, 2
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
In2
Out2
In1
Out1
Eric Pacuit 71
Rationalization versus Mistakes
A3, 3
A1, 1
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
In2
Out2
In1
Out1
Eric Pacuit 71
Rationalization versus Mistakes
A3, 3
A4, 4
B
A
l ru 4, 1 0, 0d 0, 0 1, 4
In2
Out2
In1
Out1
Eric Pacuit 71
Allowing for mistakes
A
A2
2 0
in1
in2
out1
out2
D1
A
2 2 0
out1 out2 in
D2
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A. Knoks and EP. Interpreting Mistakes in Games: From Beliefs about Mistakes to Mis-taken Beliefs. manuscript, 2016.
Eric Pacuit 73
Our Model
MG = 〈W , {(βi , σi)}i∈N, {�i}i∈N, {Pi}i∈N〉
A : Ou; B : c
A : Iu; B : c
The players’ explanation/prediction
The observed behavior
I two functions: β and σI states represent ex interim stages of the
gameI what is a “mistake”?
Eric Pacuit 74
Our Model
MG = 〈W , {(βi , σi)}i∈N, {�i}i∈N, {Pi}i∈N〉
A : Ou; B : c
A : Iu; B : c
The players’ explanation/prediction
The observed behavior
I two functions: β and σI states represent ex interim stages of the
gameI what is a “mistake”?
Eric Pacuit 74
Our Model
MG = 〈W , {(βi , σi)}i∈N, {�i}i∈N, {Pi}i∈N〉
A : Ou; B : c
A : Iu; B : c
The players’ explanation/prediction
The observed behavior
I two functions: β and σ
I states represent ex interim stages of thegame
I what is a “mistake”?
Eric Pacuit 74
Our Model
MG = 〈W , {(βi , σi)}i∈N, {�i}i∈N, {Pi}i∈N〉
A : Ou; B : c
A : Iu; B : c
The players’ explanation/prediction
The observed behavior
I two functions: β and σI states represent ex interim stages of the
game
I what is a “mistake”?
Eric Pacuit 74
Our Model
MG = 〈W , {(βi , σi)}i∈N, {�i}i∈N, {Pi}i∈N〉
A : Ou; B : c
A : Iu; B : c
The players’ explanation/prediction
The observed behavior
I two functions: β and σI states represent ex interim stages of the
gameI what is a “mistake”?
Eric Pacuit 74
Suppose that W is a nonempty set of states. Each player i will beassociated with two functions βi and σi subject to the followingconstraints:
1. For each i ∈ N, βi(w) is a (possibly empty) i-history and σi(w) is astrategy for player i.
2. The i-histories {βi(w)}i∈N are coherent.
A player made a mistake at a history h ∈ Vi in w provided thatβi(w)h , σi(w)(h) (if βi(w)h is defined).
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A ,B
v1
2, 2 2, 0 2, 1 1, 1 1, 5 4, 0
A
v0
3, 3
I
O u, a u, c d, a d, cu, b d, b
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A :Ou; B:a
A :I; B:c
w1
A :Ou; B:b
A :I; B:c
w2
A :Ou; B:c
A :I; B:c
w3
Eric Pacuit 77
For a game G, a game model is a tupleMG = 〈W , {(βi , σi)}i∈N , {�i}i∈N , {Pi}i∈N〉, where:
I For all w ∈ W and i ∈ N, if v ∈ [w]i , then σi(w) = σi(v). That is,players know their own strategy.
I For all w ∈ W and i ∈ N, for each initial segment h′ ⊆ hw (includingthe empty history), there is a w′ ∈ [w]i such that hw = h′.
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The Model: Example
A :Ou; B:a
A :Iu ; B:aw4
A :Ou; B:a
A :I ; B:w3
A :Ou; B:a
A : ; B:w1
A :Od; B:a
A : ; B:w2
A ,B h2A
h1
2, 2 2, 0 2, 1
1, 1 1, 5 4, 03, 3
I
O
a c
a c
b
b
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Beliefs
[w]i = {v | w �i v or v �i w}
Pi,w(E) = Pi(E | max�i
([w]i))
Given any evidence F ⊆ W :
Pi,w(E | F) = Pi(E | max�i
(F ∩ [w]i)).
Eric Pacuit 80
Beliefs
[w]i = {v | w �i v or v �i w}
Pi,w(E) = Pi(E | max�i
([w]i))
Given any evidence F ⊆ W :
Pi,w(E | F) = Pi(E | max�i
(F ∩ [w]i)).
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Each state w ∈ W is associated with a history hw corresponding to theplay of game associated with {βi(w)}i∈N.
Note that hw need not be a maximal history, so hw is the behavior that isobserved at state w.
For any h ∈ H, let [h] = {w | βi(w) = behi(h) for all i ∈ N} be the eventthat the players behaved according to history h.
Pi,w(E | [hw ]) is i’s probability of E given her most plausible explanationof the actions she observed at state w.
Eric Pacuit 81
Each state w ∈ W is associated with a history hw corresponding to theplay of game associated with {βi(w)}i∈N.
Note that hw need not be a maximal history, so hw is the behavior that isobserved at state w.
For any h ∈ H, let [h] = {w | βi(w) = behi(h) for all i ∈ N} be the eventthat the players behaved according to history h.
Pi,w(E | [hw ]) is i’s probability of E given her most plausible explanationof the actions she observed at state w.
Eric Pacuit 81
Each state w ∈ W is associated with a history hw corresponding to theplay of game associated with {βi(w)}i∈N.
Note that hw need not be a maximal history, so hw is the behavior that isobserved at state w.
For any h ∈ H, let [h] = {w | βi(w) = behi(h) for all i ∈ N} be the eventthat the players behaved according to history h.
Pi,w(E | [hw ]) is i’s probability of E given her most plausible explanationof the actions she observed at state w.
Eric Pacuit 81
Each state w ∈ W is associated with a history hw corresponding to theplay of game associated with {βi(w)}i∈N.
Note that hw need not be a maximal history, so hw is the behavior that isobserved at state w.
For any h ∈ H, let [h] = {w | βi(w) = behi(h) for all i ∈ N} be the eventthat the players behaved according to history h.
Pi,w(E | [hw ]) is i’s probability of E given her most plausible explanationof the actions she observed at state w.
Eric Pacuit 81
Optimal Choice - Induced Strategy
For each w ∈ W , the strategy realized at w by player i issi(w) : Vi → Acti defined as follows:
si(w)(h) =
βi(w)h if βi(w)h is defined
σi(w)(h) otherwise
Then, s(w) = (s1(w), . . . , sn(w)) is a profile of strategies, and let Out(s)be the (unique) terminal history generated by s.
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Optimal Choice - Expected Utility
For any strategy si ∈ Si for player i, the expected utility of si at state wis:
EUi,w(si) =∑
w′∈W
Pi,w({w′} | [hw ])ui(Out(si , s−i(w))).
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Optimal Choice
Let Si(w) ⊆ Si be the set of strategies for player i that conform to playeri’s moves in state w.
Opti = {w | σi(w) maximizes expected utilitywith respect to Pi,w and Si(w)}.
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Rational Choice
We say that a state w′ ∈ [w]i is an earlier choice state provided βi(w′)is an initial segment of βi(w).
Player i is rational-1 at state w provided w′ ∈ Opti for all earlier choicestates w′. Let Rat1
i be the set of all states w such that i is rational-1 in w.
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What can we do with our model?
I Represent FI, BI, as well as “hybrid” belief revision policies
I Condition on choices, “mistakes”, observed behavior,rationality, etc.
I Prove (or re-prove) “characterization results”
I An implicit assumption in EGT literature: choice of strategy impliesits execution: Our framework highlights the role that this assumptionplays in (epistemic) game-theoretic analyses
Eric Pacuit 86
What can we do with our model?
I Represent FI, BI, as well as “hybrid” belief revision policies
I Condition on choices, “mistakes”, observed behavior,rationality, etc.
I Prove (or re-prove) “characterization results”
I An implicit assumption in EGT literature: choice of strategy impliesits execution: Our framework highlights the role that this assumptionplays in (epistemic) game-theoretic analyses
Eric Pacuit 86
What can we do with our model?
I Represent FI, BI, as well as “hybrid” belief revision policies
I Condition on choices, “mistakes”, observed behavior,rationality, etc.
I Prove (or re-prove) “characterization results”
I An implicit assumption in EGT literature: choice of strategy impliesits execution: Our framework highlights the role that this assumptionplays in (epistemic) game-theoretic analyses
Eric Pacuit 86
What can we do with our model?
I Represent FI, BI, as well as “hybrid” belief revision policies
I Condition on choices, “mistakes”, observed behavior,rationality, etc.
I Prove (or re-prove) “characterization results”
I An implicit assumption in EGT literature: choice of strategy impliesits execution: Our framework highlights the role that this assumptionplays in (epistemic) game-theoretic analyses
Eric Pacuit 86
What can we do with our model?
I Represent FI, BI, as well as “hybrid” belief revision policies
I Condition on choices, “mistakes”, observed behavior,rationality, etc.
I Prove (or re-prove) “characterization results”
I An implicit assumption in EGT literature: choice of strategy impliesits execution: Our framework highlights the role that this assumptionplays in (epistemic) game-theoretic analyses
Eric Pacuit 86
Concluding Remarks: Irrationality in Games
[T]he rules of rational behavior must provide definitely for thepossibility of irrational conduct on the part of others. Imaginethat we have discovered a set of rules for all participants—to betermed as “optimal” or “rational”—each of which is indeedoptimal provided that the other participants conform. Then thequestion remains as to what will happen if some of theparticipants do not conform.adfs (von Neumann & Morgenstern, pg. 32)
Eric Pacuit 87
Concluding Remarks: Richness ConditionsMany epistemic characterization results make a richness assumptionabout the epistemic models.
I What is a “good” epistemic characterization result?
X Players need “enough” conditional beliefs to “make sense of”observed behavior.
P. Battigalli and M. Siniscalchi. Strong Belief and Forward Induction Reasoning. Journalof Economic Theory 106(2), pgs. 356391, 2002.
R. Stalnaker. Belief revision in games: forward and backward induction. MathematicalSocial Sciences, 36, pgs. 31 - 56, 1998.
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Concluding Remarks: Reasoning in Games
“The word eductive will be used to describe a dynamic process by meansof which equilibrium is achieved through careful reasoning on the part ofthe players. Such reasoning will usually require an attempt to simulatethe reasoning processes of the other players. Some measure of pre-playcommunication is therefore implied, although this need not be explicit. Toreason along the lines “if I think that he thinks that I think...” requires thatinformation be available on how an opponent thinks.”ad (pg. 184)
K. Binmore. Modeling Rational Players. Economics and Philosophy, 3,179 - 21, 1987.
Eric Pacuit 89
Concluding Remarks: Reasoning in Games
“The fundamental insight of game theory [is] that a rational player musttake into account that the players reason about each other in decidinghow to play” (pg. 81)
R. Aumann and J. Dreze. Rational expectations in games. American Economic Review,Vol. 98, pgs. 72 – 86 (2008).
Exactly how the players incorporate the fact that they are interacting withother (actively reasoning) rational agents is the subject of much debate.
Eric Pacuit 90
Concluding Remarks: Reasoning in Games
“The fundamental insight of game theory [is] that a rational player musttake into account that the players reason about each other in decidinghow to play” (pg. 81)
R. Aumann and J. Dreze. Rational expectations in games. American Economic Review,Vol. 98, pgs. 72 – 86 (2008).
Exactly how the players incorporate the fact that they are interacting withother (actively reasoning) rational agents is the subject of much debate.
Eric Pacuit 90
Thank You!
Eric Pacuit 91