Topics in Algorithmic Game Theory and Economics
Pieter Kleer
Max Planck Institute for Informatics Saarland Informatics Campus
December 16, 2020
Lecture 6
Finite games III - Computation of CE and CCE
Hierarchy of equilibrium concepts
Finite (cost minimization) gameΓ = (N,(Si)i∈N,(Ci)i∈N)consists of:
Finite setN ofplayers.
Finitestrategy setSi for every playeri ∈N. Cost functionCi :×jSj →Rfor everyi∈N.
PNE
Exists in any congestion game MNE
Exists in any finite game, but hard to compute CE
Computationally tractable CCE
Hierarchy of equilibrium concepts
Finite (cost minimization) gameΓ = (N,(Si)i∈N,(Ci)i∈N)consists of:
Finite setN ofplayers.
Finitestrategy setSi for every playeri ∈N. Cost functionCi :×jSj →Rfor everyi∈N.
PNE
Exists in any congestion game MNE
Exists in any finite game, but hard to compute CE
Computationally tractable CCE
Hierarchy of equilibrium concepts
Finite (cost minimization) gameΓ = (N,(Si)i∈N,(Ci)i∈N)consists of:
Finite setN ofplayers.
Finitestrategy setSifor every playeri ∈N.
Cost functionCi :×jSj →Rfor everyi∈N.
PNE
Exists in any congestion game MNE
Exists in any finite game, but hard to compute CE
Computationally tractable CCE
Hierarchy of equilibrium concepts
Finite (cost minimization) gameΓ = (N,(Si)i∈N,(Ci)i∈N)consists of:
Finite setN ofplayers.
Finitestrategy setSifor every playeri ∈N.
Cost functionCi :×jSj →Rfor everyi∈N.
PNE
Exists in any congestion game MNE
Exists in any finite game, but hard to compute CE
Computationally tractable CCE
Hierarchy of equilibrium concepts
Finite (cost minimization) gameΓ = (N,(Si)i∈N,(Ci)i∈N)consists of:
Finite setN ofplayers.
Finitestrategy setSifor every playeri ∈N.
Cost functionCi :×jSj →Rfor everyi∈N.
MNE CE
Computationally tractable CCE
Two-player games with mixed strategies (recap)
Two-player game(A,B)given by matricesA,B ∈Rm×n.
Row player Alice and column player Bobindependentlychoose strategyx ∈∆Aandy ∈∆B.
Givesproduct distributionσx,y :SA× SB →[0,1]over strategy profiles: σx,y(ak,b`) =σk` =xky` fork =1, . . . ,mand`=1, . . . ,n.
Example
Distribution over strategy profiles is given by x1y1 x1y2 x1y3 x2y1 x2y2 x2y3
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Thenexpected cost(for Alice)CA(σx,y)is xTAy =E(ak,b`)∼σx,y[CA(ak,b`)] = X
(ak,b`)∈SA×SB
σk`CA(ak,b`)
Remember thatAk`=CA(ak,b`).
Two-player games with mixed strategies (recap)
Two-player game(A,B)given by matricesA,B ∈Rm×n.
Row player Alice and column player Bobindependentlychoose strategyx ∈∆Aandy ∈∆B.
Givesproduct distributionσx,y :SA× SB →[0,1]over strategy profiles:
σx,y(ak,b`) =σk` =xky` fork =1, . . . ,mand`=1, . . . ,n. Example
Distribution over strategy profiles is given by x1y1 x1y2 x1y3 x2y1 x2y2 x2y3
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Thenexpected cost(for Alice)CA(σx,y)is xTAy =E(ak,b`)∼σx,y[CA(ak,b`)] = X
(ak,b`)∈SA×SB
σk`CA(ak,b`)
Remember thatAk`=CA(ak,b`).
Two-player games with mixed strategies (recap)
Two-player game(A,B)given by matricesA,B ∈Rm×n.
Row player Alice and column player Bobindependentlychoose strategyx ∈∆Aandy ∈∆B.
Givesproduct distributionσx,y :SA× SB →[0,1]over strategy profiles:
σx,y(ak,b`) =σk` =xky` fork =1, . . . ,mand`=1, . . . ,n.
Example
Distribution over strategy profiles is given by x1y1 x1y2 x1y3 x2y1 x2y2 x2y3
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Thenexpected cost(for Alice)CA(σx,y)is xTAy =E(ak,b`)∼σx,y[CA(ak,b`)] = X
(ak,b`)∈SA×SB
σk`CA(ak,b`)
Remember thatAk`=CA(ak,b`).
Two-player games with mixed strategies (recap)
Two-player game(A,B)given by matricesA,B ∈Rm×n.
Row player Alice and column player Bobindependentlychoose strategyx ∈∆Aandy ∈∆B.
Givesproduct distributionσx,y :SA× SB →[0,1]over strategy profiles:
σx,y(ak,b`) =σk` =xky` fork =1, . . . ,mand`=1, . . . ,n.
Example
Distribution over strategy profiles is given by x1y1 x1y2 x1y3 x2y1 x2y2 x2y3
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Thenexpected cost(for Alice)CA(σx,y)is xTAy =E(ak,b`)∼σx,y[CA(ak,b`)] = X
(ak,b`)∈SA×SB
σk`CA(ak,b`)
Remember thatAk`=CA(ak,b`).
Two-player games with mixed strategies (recap)
Two-player game(A,B)given by matricesA,B ∈Rm×n.
Row player Alice and column player Bobindependentlychoose strategyx ∈∆Aandy ∈∆B.
Givesproduct distributionσx,y :SA× SB →[0,1]over strategy profiles:
σx,y(ak,b`) =σk` =xky` fork =1, . . . ,mand`=1, . . . ,n.
Example
Distribution over strategy profiles is given by x1y1 x1y2 x1y3 x2y1 x2y2 x2y3
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Thenexpected cost(for Alice)CA(σx,y)is
xTAy =E(ak,b`)∼σx,y[CA(ak,b`)] = X
(ak,b`)∈SA×SB
σk`CA(ak,b`)
Remember thatAk`=CA(ak,b`).
Two-player games with mixed strategies (recap)
Two-player game(A,B)given by matricesA,B ∈Rm×n.
Row player Alice and column player Bobindependentlychoose strategyx ∈∆Aandy ∈∆B.
Givesproduct distributionσx,y :SA× SB →[0,1]over strategy profiles:
σx,y(ak,b`) =σk` =xky` fork =1, . . . ,mand`=1, . . . ,n.
Example
Distribution over strategy profiles is given by x1y1 x1y2 x1y3 x2y1 x2y2 x2y3
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Thenexpected cost(for Alice)CA(σx,y)is
Remember thatAk`=CA(ak,b`).
Two-player games with mixed strategies (recap)
Two-player game(A,B)given by matricesA,B ∈Rm×n.
Row player Alice and column player Bobindependentlychoose strategyx ∈∆Aandy ∈∆B.
Givesproduct distributionσx,y :SA× SB →[0,1]over strategy profiles:
σx,y(ak,b`) =σk` =xky` fork =1, . . . ,mand`=1, . . . ,n.
Example
Distribution over strategy profiles is given by x1y1 x1y2 x1y3 x2y1 x2y2 x2y3
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Thenexpected cost(for Alice)CA(σx,y)is xTAy =E(ak,b`)∼σx,y[CA(ak,b`)] = X
σk`CA(ak,b`)
Beyond mixed strategies
Equilibrium concepts as distributions over S
A× S
BWe have seen the following equilibrium concepts:
PNE: Strategy profiles = (sA,sB)∈ SA× SB. Givesindicator distributionσoverSA× SBwith
σ(t) =
1 t =s 0 t 6=s . MNE: Mixed strategies(x,y)withx ∈∆A,y ∈∆B.
Givesproduct distributionσoverSA× SB, where σ(ak,b`) =σk`=xky`.
(C)CE: (Coarse) correlated equilibrium will be defined as general distributionσ overSA× SB.
I.e., not induced by specific player actions.
Equilibrium concepts as distributions over S
A× S
BWe have seen the following equilibrium concepts:
PNE: Strategy profiles= (sA,sB)∈ SA× SB.
Givesindicator distributionσoverSA× SBwith σ(t) =
1 t =s 0 t 6=s . MNE: Mixed strategies(x,y)withx ∈∆A,y ∈∆B.
Givesproduct distributionσoverSA× SB, where σ(ak,b`) =σk`=xky`.
(C)CE: (Coarse) correlated equilibrium will be defined as general distributionσ overSA× SB.
I.e., not induced by specific player actions.
Equilibrium concepts as distributions over S
A× S
BWe have seen the following equilibrium concepts:
PNE: Strategy profiles= (sA,sB)∈ SA× SB. Givesindicator distributionσoverSA× SBwith
σ(t) =
1 t =s 0 t 6=s .
MNE: Mixed strategies(x,y)withx ∈∆A,y ∈∆B. Givesproduct distributionσoverSA× SB, where
σ(ak,b`) =σk`=xky`.
(C)CE: (Coarse) correlated equilibrium will be defined as general distributionσ overSA× SB.
I.e., not induced by specific player actions.
Equilibrium concepts as distributions over S
A× S
BWe have seen the following equilibrium concepts:
PNE: Strategy profiles= (sA,sB)∈ SA× SB. Givesindicator distributionσoverSA× SBwith
σ(t) =
1 t =s 0 t 6=s . MNE: Mixed strategies(x,y)withx ∈∆A,y ∈∆B.
Givesproduct distributionσoverSA× SB, where σ(ak,b`) =σk`=xky`.
(C)CE: (Coarse) correlated equilibrium will be defined as general distributionσ overSA× SB.
I.e., not induced by specific player actions.
Equilibrium concepts as distributions over S
A× S
BWe have seen the following equilibrium concepts:
PNE: Strategy profiles= (sA,sB)∈ SA× SB. Givesindicator distributionσoverSA× SBwith
σ(t) =
1 t =s 0 t 6=s . MNE: Mixed strategies(x,y)withx ∈∆A,y ∈∆B.
Givesproduct distributionσoverSA× SB, where σ(ak,b`) =σk`=xky`.
(C)CE: (Coarse) correlated equilibrium will be defined as general distributionσ overSA× SB.
I.e., not induced by specific player actions.
Equilibrium concepts as distributions over S
A× S
BWe have seen the following equilibrium concepts:
PNE: Strategy profiles= (sA,sB)∈ SA× SB. Givesindicator distributionσoverSA× SBwith
σ(t) =
1 t =s 0 t 6=s . MNE: Mixed strategies(x,y)withx ∈∆A,y ∈∆B.
Givesproduct distributionσoverSA× SB, where σ(ak,b`) =σk`=xky`.
(C)CE: (Coarse) correlated equilibrium will be defined as general distributionσ overSA× SB.
I.e., not induced by specific player actions.
Equilibrium concepts as distributions over S
A× S
BWe have seen the following equilibrium concepts:
PNE: Strategy profiles= (sA,sB)∈ SA× SB. Givesindicator distributionσoverSA× SBwith
σ(t) =
1 t =s 0 t 6=s . MNE: Mixed strategies(x,y)withx ∈∆A,y ∈∆B.
Givesproduct distributionσoverSA× SB, where σ(ak,b`) =σk`=xky`.
(C)CE: (Coarse) correlated equilibrium will be defined as general distributionσ overSA× SB.
I.e., not induced by specific player actions.
Game of Chicken
Game of Chicken
Alice and Bob both approach an intersection.
Bob
Stop Go
Alice Stop (0,0) (3,−1) Go (−1,3) (4,4)
Two PNEs: (Stop, Go), (Go, Stop).
One MNE: Both players randomize over Stop and Go. Distributions over strategy profiles(a,b)for these equilibria are
0 1 0 0
,
0 0 1 0
and 1
4 1 1 4 4
1 4
.
Game of Chicken
Game of Chicken
Alice and Bob both approach an intersection.
Bob
Stop Go
Alice Stop (0,0) (3,−1) Go (−1,3) (4,4) Two PNEs: (Stop, Go), (Go, Stop).
One MNE: Both players randomize over Stop and Go. Distributions over strategy profiles(a,b)for these equilibria are
0 1 0 0
,
0 0 1 0
and 1
4 1 1 4 4
1 4
.
Game of Chicken
Game of Chicken
Alice and Bob both approach an intersection.
Bob
Stop Go
Alice Stop (0,0) (3,−1) Go (−1,3) (4,4) Two PNEs: (Stop, Go), (Go, Stop).
One MNE: Both players randomize over Stop and Go.
Distributions over strategy profiles(a,b)for these equilibria are 0 1
0 0
,
0 0 1 0
and 1
4 1 1 4 4
1 4
.
Game of Chicken
Game of Chicken
Alice and Bob both approach an intersection.
Bob
Stop Go
Alice Stop (0,0) (3,−1) Go (−1,3) (4,4) Two PNEs: (Stop, Go), (Go, Stop).
One MNE: Both players randomize over Stop and Go.
Distributions over strategy profiles(a,b)for these equilibria are 0 1
0 0
,
0 0 1 0
and 1
4 1 1 4 4
1 4
.
Game of Chicken
Game of Chicken
Alice and Bob both approach an intersection.
Bob
Stop Go
Alice Stop (0,0) (3,−1) Go (−1,3) (4,4) Two PNEs: (Stop, Go), (Go, Stop).
One MNE: Both players randomize over Stop and Go.
Distributions over strategy profiles(a,b)for these equilibria are
Sensible ‘equilibrium’ would be the strategy profile distribution
σ =
0 12
1 2 0
.
Cannot be achieved as mixed equilibrium.
There arenox ∈∆A,y ∈∆Bsuch thatσk`=xky`for all k, `∈ {1,2}.
Idea is to introducetraffic light(mediator or trusted third party). Traffic light samples/draws one of the two strategy profiles from distribution.
Gives realization as recommendation to the players. Tells Alice to go and Bob to stop (or vice versa)
Conditioned on this recommendation, the best thing for a player to do is to follow it.
Sensible ‘equilibrium’ would be the strategy profile distribution
σ =
0 12
1 2 0
.
Cannot be achieved as mixed equilibrium.
There arenox ∈∆A,y ∈∆Bsuch thatσk`=xky`for all k, `∈ {1,2}.
Idea is to introducetraffic light(mediator or trusted third party). Traffic light samples/draws one of the two strategy profiles from distribution.
Gives realization as recommendation to the players. Tells Alice to go and Bob to stop (or vice versa)
Conditioned on this recommendation, the best thing for a player to do is to follow it.
Sensible ‘equilibrium’ would be the strategy profile distribution
σ =
0 12
1 2 0
.
Cannot be achieved as mixed equilibrium.
There arenox ∈∆A,y ∈∆Bsuch thatσk`=xky`for all k, `∈ {1,2}.
Idea is to introducetraffic light(mediator or trusted third party). Traffic light samples/draws one of the two strategy profiles from distribution.
Gives realization as recommendation to the players. Tells Alice to go and Bob to stop (or vice versa)
Conditioned on this recommendation, the best thing for a player to do is to follow it.
Sensible ‘equilibrium’ would be the strategy profile distribution
σ =
0 12
1 2 0
.
Cannot be achieved as mixed equilibrium.
There arenox ∈∆A,y ∈∆Bsuch thatσk`=xky`for all k, `∈ {1,2}.
Idea is to introducetraffic light(mediator or trusted third party).
Traffic light samples/draws one of the two strategy profiles from distribution.
Gives realization as recommendation to the players. Tells Alice to go and Bob to stop (or vice versa)
Conditioned on this recommendation, the best thing for a player to do is to follow it.
Sensible ‘equilibrium’ would be the strategy profile distribution
σ =
0 12
1 2 0
.
Cannot be achieved as mixed equilibrium.
There arenox ∈∆A,y ∈∆Bsuch thatσk`=xky`for all k, `∈ {1,2}.
Idea is to introducetraffic light(mediator or trusted third party).
Traffic light samples/draws one of the two strategy profiles from distribution.
Gives realization as recommendation to the players. Tells Alice to go and Bob to stop (or vice versa)
Conditioned on this recommendation, the best thing for a player to do is to follow it.
Sensible ‘equilibrium’ would be the strategy profile distribution
σ =
0 12
1 2 0
.
Cannot be achieved as mixed equilibrium.
There arenox ∈∆A,y ∈∆Bsuch thatσk`=xky`for all k, `∈ {1,2}.
Idea is to introducetraffic light(mediator or trusted third party).
Traffic light samples/draws one of the two strategy profiles from distribution.
Gives realization as recommendation to the players.
Tells Alice to go and Bob to stop (or vice versa)
Conditioned on this recommendation, the best thing for a player to do is to follow it.
Sensible ‘equilibrium’ would be the strategy profile distribution
σ =
0 12
1 2 0
.
Cannot be achieved as mixed equilibrium.
There arenox ∈∆A,y ∈∆Bsuch thatσk`=xky`for all k, `∈ {1,2}.
Idea is to introducetraffic light(mediator or trusted third party).
Traffic light samples/draws one of the two strategy profiles from distribution.
Gives realization as recommendation to the players.
Tells Alice to go and Bob to stop (or vice versa)
Conditioned on this recommendation, the best thing for a player to do is to follow it.
Sensible ‘equilibrium’ would be the strategy profile distribution
σ =
0 12
1 2 0
.
Cannot be achieved as mixed equilibrium.
There arenox ∈∆A,y ∈∆Bsuch thatσk`=xky`for all k, `∈ {1,2}.
Idea is to introducetraffic light(mediator or trusted third party).
Traffic light samples/draws one of the two strategy profiles from distribution.
Gives realization as recommendation to the players.
Tells Alice to go and Bob to stop (or vice versa)
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows. Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob. Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn. Player assumes all other players play private recommendation, i.e., Alice
assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ. σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob. Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn. Player assumes all other players play private recommendation, i.e., Alice
assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob. Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn. Player assumes all other players play private recommendation, i.e., Alice
assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob. Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn. Player assumes all other players play private recommendation, i.e., Alice
assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation! Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn. Player assumes all other players play private recommendation, i.e., Alice
assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn. Player assumes all other players play private recommendation, i.e., Alice
assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn. Player assumes all other players play private recommendation, i.e., Alice
assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn.
Player assumes all other players play private recommendation, i.e., Alice assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn.
Player assumes all other players play private recommendation,
i.e., Alice assumes Bob follows his recommendation (and vice versa). In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn.
Player assumes all other players play private recommendation, i.e., Alice assumes Bob follows his recommendation (and vice versa).
In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn.
Player assumes all other players play private recommendation, i.e., Alice assumes Bob follows his recommendation (and vice versa).
In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE). Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn.
Player assumes all other players play private recommendation, i.e., Alice assumes Bob follows his recommendation (and vice versa).
In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse
Therefore, for (C)CE, it’s not always necessary that all players know the distributionσ up front, nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn.
Player assumes all other players play private recommendation, i.e., Alice assumes Bob follows his recommendation (and vice versa).
In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse correlated equilibrium (similar algorithms exist converging to CE).
Therefore, for (C)CE, it’s not always necessary that all players know
nor that there is an actual third party that samples from it.
Correlated equilibrium (CE), informal
Correlated equilibriumσ :SA× SB →[0,1]can be seen as follows.
Mediator (third party) draws samplex = (xA,xB)∼σ.
σis known to Alice and Bob, but notx.
Givesprivate recommendationxAto Alice, andxB to Bob.
Alice and Bob do not know each other’s recommendation!
Game of Chicken is the exception to the rule.
Recommendations give players some info on whichx was drawn.
Player assumes all other players play private recommendation, i.e., Alice assumes Bob follows his recommendation (and vice versa).
In CE, no player has incentive to deviate given its recommendation.
Remark
We will later seeno-regretalgorithms whose output is a coarse
Example
Distribution over strategy profiles is given by
σ =
σ11 σ12 σ13 σ21 σ22 σ23
=
0 1/8 1/8
2/8 1/8 3/8
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Suppose Alice getssecond rowa2as recommendation.
This gives Alice a (conditional) probability distributionρfor column privately recommended to Bob:
Columnb1with probability
2 8 2
8+18+38 = 26. Columnb2with probability
1 8 2 8+1
8+3
8
= 16. Columnb3with probability
3 8 2
8+18+38 = 36.
Assuming distributionρover Bob’s recommendation, notion of CE says Alice should have no incentive to deviate to first rowa1(in expectation).
Eρ[Rowa2] =3×2/6+0×1/6+1×3/6=9/6. Eρ[Rowa1] =0×2/6+1×1/6+2×3/6=7/6.
σas above is not a CE!
Example
Distribution over strategy profiles is given by
σ =
σ11 σ12 σ13 σ21 σ22 σ23
=
0 1/8 1/8
2/8 1/8 3/8
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Suppose Alice getssecond rowa2as recommendation.
This gives Alice a (conditional) probability distributionρfor column privately recommended to Bob:
Columnb1with probability
2 8 2
8+18+38 = 26. Columnb2with probability
1 8 2 8+1
8+3
8
= 16. Columnb3with probability
3 8 2
8+18+38 = 36.
Assuming distributionρover Bob’s recommendation, notion of CE says Alice should have no incentive to deviate to first rowa1(in expectation).
Eρ[Rowa2] =3×2/6+0×1/6+1×3/6=9/6. Eρ[Rowa1] =0×2/6+1×1/6+2×3/6=7/6.
σas above is not a CE!
Example
Distribution over strategy profiles is given by
σ =
σ11 σ12 σ13 σ21 σ22 σ23
=
0 1/8 1/8
2/8 1/8 3/8
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Suppose Alice getssecond rowa2as recommendation.
This gives Alice a (conditional) probability distributionρfor column privately recommended to Bob:
Columnb1with probability
2 8 2
8+18+38 = 26. Columnb2with probability
1 8 2 8+1
8+3
8
= 16. Columnb3with probability
3 8 2
8+18+38 = 36.
Assuming distributionρover Bob’s recommendation, notion of CE says Alice should have no incentive to deviate to first rowa1(in expectation).
Eρ[Rowa2] =3×2/6+0×1/6+1×3/6=9/6. Eρ[Rowa1] =0×2/6+1×1/6+2×3/6=7/6.
σas above is not a CE!
Example
Distribution over strategy profiles is given by
σ =
σ11 σ12 σ13 σ21 σ22 σ23
=
0 1/8 1/8
2/8 1/8 3/8
b1 b2 b3
a1 (0,2) (1,0) (2,1) a2 (3,0) (0,1) (1,4)
Suppose Alice getssecond rowa2as recommendation.
This gives Alice a (conditional) probability distributionρfor column privately recommended to Bob:
Columnb1with probability
2 8 2
8+18+38 = 26.
Columnb2with probability
1 8 2 8+1
8+3
8
= 16. Columnb3with probability
3 8 2
8+18+38 = 36.
Assuming distributionρover Bob’s recommendation, notion of CE says Alice should have no incentive to deviate to first rowa1(in expectation).
Eρ[Rowa2] =3×2/6+0×1/6+1×3/6=9/6. Eρ[Rowa1] =0×2/6+1×1/6+2×3/6=7/6.
σas above is not a CE!
