Coalgebra of market games

Coalgebra of
market games
D. Pavlovic
Games
Coalgebra of market games
Equilibria
Positions
Conclusion
Dusko Pavlovic
Kestrel Institute
and
Oxford University
CALCO, Udine
September 2009
Outline
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Coalgebra of games
Positions
Conclusion
Equilibrium programming
Position analysis
Conclusion
Outline
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Coalgebra of games
Positions
Conclusion
Equilibrium programming
Position analysis
Conclusion
A process is a coalgebra
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
X
R
/ A ⇒ M(B × X )
Positions
Conclusion
where
◮
A — controls
◮
B — measurements
◮
X — states
A process is a coalgebra
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
A×X
R
/ M(B × X )
Positions
Conclusion
where
◮
A — controls
◮
B — measurements
◮
X — states
A process is a coalgebra
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
A×X
R
/B×X
Positions
Conclusion
where
◮
A — controls
◮
B — measurements
◮
X — states
Coalgebra of
market games
Feedback and stable control
D. Pavlovic
Games
Equilibria
A×X
R
/B×X
B×X
φ
/A
Positions
Conclusion
A×X
X
R
γ=FixA (φ◦R)
hγ,idi _
A×X
φ
/B×X
R
/A
/A
O
_φ
/B×X
Coalgebra of
market games
A game is a coalgebra
D. Pavlovic
Games
Equilibria
A×X
̺
/B×X
Positions
Conclusion
where
◮
A=
◮
B=
◮
X=
◮
n = {0, 1, . . . , n − 1} — players
Q
i∈n Ai
Q
i∈n Bi
Q
i∈n Xi
— moves
— values
— positions
Coalgebra of
market games
A game is a coalgebra
D. Pavlovic
Games
Equilibria
A×X
̺
/B×X
Positions
Conclusion
where
◮
A=
◮
B=
◮
X=
◮
n = {0, 1, . . . , n − 1} — players
Q
i∈n Ai
Q
i∈n Bi
Q
i∈n Xi
— moves
— values (ordered ring)
— positions
Response strategy and equilibrium
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
̺
A×X
/B×X
Positions
Conclusion
RS
i
A−i × Xi
/ Ai
RS=hRS i ◦πi ii∈m
A×X
RS • =Fix
A (RS)
X PPPP
PPP
PP
hRS• ,idi PPP'
A×X
/
o7 A
ooo
/
o
o
ooo
ooo RS
where
A−i
=
/A
Y
k ∈n
k ,i
Ak
Tasks for coalgebra of games
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
◮
analyze positions as states in X
Positions
Conclusion
◮
coalgebra homomorphisms between games
◮
position bisimilarity
Coalgebra of
market games
Tasks for coalgebra of games
D. Pavlovic
Games
Equilibria
◮
Positions
analyze positions as states in X
Conclusion
◮
◮
coalgebra homomorphisms between games
◮
position bisimilarity
construct equilibria as fixed points in A
RS •
/ A or position-wise X
◮
static 1
◮
equilibrium at a stationary position 1
RS •
/A
hRS • ,xi
/ A×X
Coalgebra of
market games
Tasks for coalgebra of games
D. Pavlovic
Games
Equilibria
◮
Positions
analyze positions as states in X
Conclusion
◮
◮
coalgebra homomorphisms between games
◮
position bisimilarity
construct equilibria as fixed points in A
RS •
/ A or position-wise X
◮
static 1
◮
equilibrium at a stationary position 1
RS •
/A
hRS • ,xi
/ A×X
Outline
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Coalgebra of games
Positions
Conclusion
Equilibrium programming
Position analysis
Conclusion
Coalgebra of
market games
Standard notions
(Static: X = 1)
D. Pavlovic
Games
A1 - Best response strategy RS = BR
Equilibria
Positions
Conclusion
s−i BRi si
⇐⇒
∀ti ∈ Ai . ̺i (ti , s−i ) ≤ ̺i (si , s−i )
Coalgebra of
market games
Standard notions
(Static: X = 1)
D. Pavlovic
Games
A1 - Best response relation
Equilibria
Positions
Conclusion
s BR t
⇐⇒
∀i ∈ n. s−i BRi ti
Coalgebra of
market games
Standard notions
(Static: X = 1)
D. Pavlovic
Games
A2 - Nash equilibrium
Equilibria
Positions
Conclusion
BR • s
⇐⇒
s BR s
⇐⇒
∀i ∈ n. s−i BRi si
Coalgebra of
market games
Standard notions
(Static: X = 1)
D. Pavlovic
Games
A3 - Rationalizable (undominated) profile
Equilibria
Positions
Conclusion
BR ∗s
⇐⇒
∃t. BR ∗ t ∧ tBRs
Standard notions
Coalgebra of
market games
D. Pavlovic
Upshot
Games
Equilibria
Positions
Nash equilibrium is
◮
a joint result of individual optimizations
◮
a social solution of a distributed problem
◮
noone can improve their gain on their own
◮
it leads beyond the "zero sum" view of the world
Conclusion
Standard notions
Coalgebra of
market games
D. Pavlovic
Games
Issues
◮
existence of equilibrium
◮
◮
Q: Is BR • empty?
equilibrium selection
◮
◮
Equilibria
Q: What if s, t ∈ BR • , and i plays si and j plays tj ?
social benefit of equilibrium
◮
Q: Is
P
i
̺iB (s) a global maximum (Pareto optimal)?
Positions
Conclusion
Standard notions
Coalgebra of
market games
D. Pavlovic
Games
Issues
◮
existence of equilibrium
◮
◮
A: Kakutani theorem
equilibrium selection
◮
◮
Equilibria
A: attractor dynamics
social benefit of equilibrium
◮
A: program the notions of response
Positions
Conclusion
Coalgebra of
market games
Example: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
D. Pavlovic
Games
For
Equilibria
◮
i ∈ 2 = {0, 1}
◮
Ai = {retreat, attack}
◮
Bi = R
the payoff A0 × A1
Positions
̺
Conclusion
/ B0 × B1 is given by
retreat
attack
0
w
2
retreat
w
2
w
w
2
w
attack
0
w
2
−c
−c
Coalgebra of
market games
Example: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
D. Pavlovic
Games
Equilibria
Positions
If c <
w
2,
the only Nash equilibrium 1
s
s = hattack, attacki
with the social gain of
X
̺(s) = w − 2c
Conclusion
/ A0 × A1 is
Example: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Positions
Conclusion
The unstable profile
r
= hretreat, retreati
would give the social gain of
X
̺(s) = w
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
B1 - Stable strategy RS = SR
Equilibria
Positions
Conclusion
s−i SRi si
⇐⇒
∀t ∈ A ∀ε > 0.
(1 − ε)̺i (ti , s−i ) + ε̺i (ti , t−i ) ≤
(1 − ε)̺i (si , s−i ) + ε̺i (si , t−i )
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
B1 - Stable strategy RS = SR
Equilibria
Positions
Conclusion
s−i SRi si
⇐⇒
∀ti ∈ Ai . ̺i (ti , s−i ) ≤ ̺i (si , s−i ) ∧
(̺i (ti , s−i ) = ̺i (si , s−i ) ⇒
∀t−i ∈ A−i . ̺i (ti , t−i ) ≤ ̺i (si , t−i ))
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
B1 - Stable response relation
Equilibria
Positions
Conclusion
s SR t
⇐⇒
∀i ∈ n. s−i SRi ti
Static strategies
(X = 1)
Coalgebra of
market games
D. Pavlovic
Games
B2 - Stable equilibrium
Equilibria
Positions
Conclusion
SR • s
⇐⇒
∀i ∈ n. s−i SRi si
⇐⇒
∀t ∈ A.∀i ∈ n. ̺i (ti , s−i ) ≤ ̺i (si , s−i ) ∧
̺i (si , t−i ) = ̺i (si , s−i ) ⇒
∀t−i ∈ A−i . ̺i (ti , t−i ) ≤ ̺i (si , t−i )
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
B3 - Stable (admissible) profile
Equilibria
Positions
Conclusion
SR ∗ s
⇐⇒
∃t.SR ∗ t ∧ tSRs
Coalgebra of
market games
Example: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
D. Pavlovic
Games
Equilibria
Positions
If c <
w
2,
the only stable profile 1
s
Conclusion
/ A0 × A1 is
s = hattack, attacki
with the social gain of
X
̺(s) = w − 2c
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
C1 - Uniform strategy RS = UR
Equilibria
Positions
Conclusion
s−i URi si
⇐⇒
s−i BRi si ∧
∀t−i ∈ A−i . si BR−i t−i ⇒ t−i BRi si
Static strategies
(X = 1)
Coalgebra of
market games
D. Pavlovic
Games
C2 - Uniform equilibrium
Equilibria
Positions
Conclusion
UR • s
⇐⇒
∀i ∈ n. s−i URi si
⇐⇒
BR • s ∧
∀i ∈ n. ∀t−i ∈ A−i . si BR−i t−i ⇒ t−i BRi si
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
C3 - Uniform profile
Equilibria
Positions
Conclusion
UR ∗ s
⇐⇒
∃t.UR ∗ t ∧ tURs
Example: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Positions
Conclusion
If c < w2 , there is no uniform equilibrium for Bird Politics
(neither pure nor mixed).
Example: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Positions
Conclusion
If c < w2 , there is no uniform equilibrium for Bird Politics
(neither pure nor mixed).
The Kakutani theorem does not apply because the
uniform response relation is not convex.
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
D1 - Constructive strategy RS = CR
Equilibria
Positions
Conclusion
s−i CRi si
⇐⇒
∀ti ∈ Ai . ̺i (si , s−i ) < ̺i (ti , s−i ) ⇒
∃t−i ∈ A−i . ̺−i (ti , t−i ) > ̺−i (ti , s−i ) ∧
̺i (ti , t−i ) < ̺i (si , s−i )
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
D2 - Constructive equilibrium
Equilibria
Positions
Conclusion
CR • s
⇐⇒
∀i.s−i CRi si
Coalgebra of
market games
Static strategies
(X = 1)
D. Pavlovic
Games
D3 - Uniform profile
Equilibria
Positions
Conclusion
CR ∗ s
⇐⇒
∃t.CR ∗ t ∧ tCRs
Example: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Positions
Conclusion
There is a constructive equilibrium, consisting of the
mixed strategies favoring the Pareto optimal solution
hretreat, retreati.
Outline
Coalgebra of
market games
D. Pavlovic
Games
Coalgebra of games
Equilibria
Positions
Coordination
Competition
Equilibrium programming
Position analysis
Problem of coordination
Problem of competition
Conclusion
Conclusion
Coalgebra of
market games
Tasks for coalgebra of games
D. Pavlovic
Games
Equilibria
◮
Positions
analyze positions as states in X
Coordination
Competition
◮
◮
coalgebra homomorphisms between games
◮
position bisimilarity
construct equilibria as fixed points in A
RS •
/ A or position-wise X
◮
static 1
◮
equilibrium at a stationary position 1
RS •
/A
hRS • ,xi
/ A×X
Conclusion
Types-as-positions: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
For
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
◮
i ∈ 2 = {0, 1}
◮
Ai = {retreat, attack}
◮
◮
Positions
◮
x0 = Prob(a1 = retreat)
x1 = Prob(a0 = retreat)
the payoff Ai × Xi
Competition
Conclusion
Bi = R
Xi = [0, 1]
◮
Coordination
̺iB
/ Bi is given by
w
2
w
+ 2c w − 2c
+
̺iB (attack, xi ) = xi
2
2
̺iB (retreat, xi ) = xi
Types-as-positions: Bird Politics
(Chicken, Missile Crisis, Prisoners’ Dilemma. . . )
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Positions
Coordination
Competition
Conclusion
[. . . ]
Example: Majority game
(static: X = 1)
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
For
Positions
Coordination
Competition
◮
i ∈ 2m + 1 = {0, 1, . . . 2m} — players
◮
Ai = {◭, ◮} — moves
◮
Bi = {0, 1} — values
the payoff A
̺
/ B is



1
̺ (s) = 

0
i
if #{j | sj = si } > m
otherwise
Conclusion
Coalgebra of
market games
Example: Majority game
(static: X = 1)
D. Pavlovic
Games
The equilibrium 1
s
Equilibria
/ A consists of the mixed strategies
Positions
Coordination
Competition
si
=
1
1
◭+ ◮
2
2
Conclusion
with the expected social gain of
E
X
2m+1
X
̺(s) = 2
k =m+1
= m+
1
2
2m+1
k
k
2m+1
2
Positions for coordination: Majority game
(with 1-step memory)
Coalgebra of
market games
D. Pavlovic
Games
For
Equilibria
◮
i ∈ 2m + 1 = {0, 1, . . . 2m} — players
◮
Ai = {◭, ◮} — moves
◮
Bi = {0, 1} — values
Xi = {◭, ◮} — positions
Positions
Coordination
Competition
◮
◮
ξ∈X =
the game A × X
̺iB (s, x)
Q
Conclusion
Xi — initialized randomly
̺
/ B × X becomes



1
= 

0
̺iX (s, x) = ^
if #{j | sj = si } > m
otherwise
such that #{j | sj = ^} > m
Positions for coordination: Majority game
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
The equilibria X
s
/ A are the coordination policies
si (x) = x
si′ (x) = ¬x
which assure the social gain of
X
̺(s) = 2m + 1
Positions
Coordination
Competition
Conclusion
Example: Minority game
(static: X = 1)
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
For
Positions
Coordination
Competition
◮
i ∈ 2m + 1 = {0, 1, . . . 2m} — players
◮
Ai = {◭, ◮} — moves
◮
Bi = {0, 1} — values
the payoff A
̺
/ B is



1
̺ (s) = 

0
i
if #{j | sj = si } ≤ m
otherwise
Conclusion
Coalgebra of
market games
Example: Minority game
(static: X = 1)
D. Pavlovic
Games
Equilibria
s
The static equilibrium 1
/ A is again
Positions
Coordination
Competition
si
=
1
1
◭+ ◮
2
2
Conclusion
with the expected social gain of
E
X
m
X
̺(s) =
k =1
=
m
2
2m+1
k
22m+1
k
Coalgebra of
market games
Example: Minority game
(static: X = 1)
D. Pavlovic
Games
Equilibria
Positions
Coordination
Competition
◮
Even with the maximal gain of ̺(s) = m,
there are m + 1 players with ̺i (s) = 0.
P
Conclusion
Coalgebra of
market games
Example: Minority game
(static: X = 1)
D. Pavlovic
Games
Equilibria
Positions
Coordination
Competition
◮
Even with the maximal gain of ̺(s) = m,
there are m + 1 players with ̺i (s) = 0.
◮
There is always a majority with an incentive
to disturb the current state.
P
Conclusion
Coalgebra of
market games
Example: Minority game
(static: X = 1)
D. Pavlovic
Games
Equilibria
Positions
Coordination
Competition
◮
Even with the maximal gain of ̺(s) = m,
there are m + 1 players with ̺i (s) = 0.
◮
There is always a majority with an incentive
to disturb the current state.
◮
This leads from equilibrium to evolution.
P
Conclusion
Positions for stabilization: Minority game
(with ℓ-step memory and d ideas)
Coalgebra of
market games
D. Pavlovic
Games
Suppose that the player i sees the positions from
Equilibria
Positions
Xi
= M ×S
1+d
×ℓ
Coordination
Competition
Conclusion
where
◮
M = S = {◭, ◮}ℓ — memory, strategies, ideas
◮
d = {0, 1, . . . , d − 1} — number of ideas
◮
ℓ = {0, 1, . . . , ℓ − 1} — length of the memory
Positions for stabilization: Minority game
(with ℓ-step memory and d ideas)
Coalgebra of
market games
D. Pavlovic
Games
A position
Equilibria
Positions
i0
i1
id
xi = hµ, σ , σ , . . . , σ , ki ∈ M × S
1+d
× ℓ = Xi
Coordination
Competition
Conclusion
records
◮
µ — the recent ℓ minority (winning) choices
◮
σi0 — i’s current strategy (ℓ-tuple of choices)
◮
σi1 , σi2 , . . . , σid — i’s bag of ideas for strategies
◮
k — the current moment in the ℓ-cycle history
Positions for stabilization: Minority game
Coalgebra of
market games
D. Pavlovic
Games
For
Equilibria
◮
i ∈ 2m + 1 = {0, 1, . . . 2m} — players
◮
Ai = {◭, ◮} — moves
◮
Bi = {0, 1} — values
Xi = M × S 1+d × ℓ — positions
Positions
Coordination
Competition
◮
◮
Conclusion
ξi ∈ Xi — initialized randomly
the payoff A × X
̺iB (s, x)
̺B
/ B remains



1
= 

0
if #{j | sj = si } ≤ m
otherwise
Positions for stabilization: Minority game
. . . while the position update A × Xi
̺iX
Coalgebra of
market games
D. Pavlovic
/ Xi maps
Games
Equilibria
̺iX (s,
i∗
i∗
hµ, σ , ki) = hµ̃, σ̃ , k̃i
Positions
Coordination
Competition
so that
Conclusion
◮
k̃ = k + 1 mod ℓ
◮
µ̃ = h^, µ0 , µ1 , . . . , µℓ−2 i
◮
◮
where ^ is the minority choice, i.e. #{j | sj = ^} ≤ m
σ̃i∗ is obtained by reordering
◮
σ̂i∗ = hσi∗
, σi∗
, σi∗
, . . . , σi∗
i
ℓ−1
0
1
ℓ−2
◮
to maintain the invariant
∆(σ̃i0 , µ̃) ≤ ∆(σ̃i1 , µ̃) ≤ · · · ≤ ∆(σ̃id , µ̃)
Positions for stabilization: Minority game
. . . while the position update A × Xi
̺iX
Coalgebra of
market games
D. Pavlovic
/ Xi maps
Games
Equilibria
̺iX (s,
i∗
i∗
hµ, σ , ki) = hµ̃, σ̃ , k̃i
Positions
Coordination
Competition
so that
Conclusion
◮
k̃ = k + 1 mod ℓ
◮
µ̃ = h^, µ0 , µ1 , . . . , µℓ−2 i
◮
◮
where ^ is the minority choice, i.e. #{j | sj = ^} ≤ m
σ̃i∗ is obtained by reordering
◮
σ̂i∗ = hσi∗
, σi∗
, σi∗
, . . . , σi∗
i
ℓ−1
0
1
ℓ−2
◮
to maintain the invariant
∆(σ̃i0 , µ̃) ≤ ∆(σ̃i1 , µ̃) ≤ · · · ≤ ∆(σ̃id , µ̃)
— thus σ̃i0 is the best and σ̃id the worst strategy w.r.t. µ̃
Positions for stabilization: Minority game
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Positions
Coordination
Competition
Let the profile X
s
/ A be defined by
si (µ, σi∗, k) = σi0
k
i.e., each player plays his currently best strategy.
Conclusion
Positions for stabilization: Minority game
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Evolution: refine A × Xi
̺iX
Positions
/
X
i
Coordination
Competition
Conclusion
◮
Each player randomly mutates her state by
◮
dropping her worst idea σi(ℓ−1) ∈ {◭, ◮}ℓ
◮
adding a random idea σ′ ∈ {◭, ◮}ℓ .
at chosen intervals, or triggered by bad scores.
Positions for stabilization: Minority game
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Evolution: refine A × Xi
̺iX
Positions
/
X
i
Coordination
Competition
Conclusion
◮
Each player randomly mutates her state by
◮
dropping her worst idea σi(ℓ−1) ∈ {◭, ◮}ℓ
◮
adding a random idea σ′ ∈ {◭, ◮}ℓ .
at chosen intervals, or triggered by bad scores.
◮
This leads to jointly stable populations of players.
Positions for stabilization: Minority game
Coalgebra of
market games
D. Pavlovic
Games
Stable positions
Equilibria
Positions
X
hs,idi
/A×X
A×X
hs,idi
̺X
X
1
/A×X
η=FixX (̺X ◦s)
η_
X
hs,idi
̺X
Coordination
/X
/X
/X
O
_ ̺X
/A×X
yield improved social gains E(Σ̺) >
m
2
+ q.
Competition
Conclusion
Coalgebra of
market games
Example: Market game
(static: X = 1)
D. Pavlovic
Games
For
Equilibria
◮
i ∈ n = {0, 1, . . . n − 1} — players (sellers, producers)
◮
Ai = Bi = R — moves, values
the payoff A
̺
/ B is



si − ci
̺ (s) = 

0
i
if ∀j∈n\{i} . si < sj
otherwise
Positions
Coordination
Competition
Conclusion
Coalgebra of
market games
Example: Market game
(static: X = 1)
D. Pavlovic
Games
For
Equilibria
◮
i ∈ n = {0, 1, . . . n − 1} — players (sellers, producers)
◮
Ai = Bi = R — moves, values
the payoff A
̺
/ B is



si − ci
̺ (s) = 

0
i
if ∀j∈n\{i} . si < sj
otherwise
where
◮
si is the market price offered by the producer i,
◮
ci is the production cost of i
Positions
Coordination
Competition
Conclusion
Example: Market game
(static: X = 1)
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Positions
Coordination
Competition
The equilibria 1
s
Conclusion
/ A consist of the strategies
si
= ci + εi
where εi ∈ [pi , qi ] is the desired profit.
Example: Market game
(with memory and tactics)
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Marketing tactics (equilibrium selection)
Positions
Coordination
Competition
◮
to win, find ε such that ci + ε < cj + εj for all j , i
Conclusion
Example: Market game
(with memory and tactics)
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Marketing tactics (equilibrium selection)
Positions
Coordination
Competition
◮
to win, find ε such that ci + ε < cj + εj for all j , i
◮
to profit, maximize among such ε
Conclusion
Example: Market game
(with memory and tactics)
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Marketing tactics (equilibrium selection)
Positions
Coordination
Competition
◮
to win, find ε such that ci + ε < cj + εj for all j , i
◮
◮
to profit, maximize among such ε
change the game:
◮
sway the buyer to pay more than the lowest price
◮
◮
lock in, bundling, price discrimination. . .
manipulate the market information
◮
advertising, branding. . .
Conclusion
Stable solution: Second price market game
(Static: X = 1)
Coalgebra of
market games
D. Pavlovic
Games
For
Equilibria
◮
i ∈ n = {0, 1, . . . n − 1} — players
◮
Ai = Bi = R — moves, values
the payoff A
̺
/ B is



⌈si ⌉s − si
̺ (s) = 

0
i
if ∀j∈n\{i} . si < sj
otherwise
Positions
Coordination
Competition
Conclusion
Stable solution: Second price market game
(Static: X = 1)
Coalgebra of
market games
D. Pavlovic
Games
For
Equilibria
◮
i ∈ n = {0, 1, . . . n − 1} — players
◮
Ai = Bi = R — moves, values
the payoff A
̺
/ B is



⌈si ⌉s − si
̺ (s) = 

0
i
if ∀j∈n\{i} . si < sj
otherwise
where
⌈a⌉β =
^
{b ∈ β | a < b}
Positions
Coordination
Competition
Conclusion
Stable solution: Second price market game
(Static: X = 1)
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
The unique equilibrium 1
strategies
s
/ A consists of the
Positions
Coordination
Competition
Conclusion
si
= ci
Stable solution: Second price market game
(Static, stable, unimplementable)
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
The unique equilibrium 1
strategies
s
/ A consists of the
Positions
Coordination
Competition
Conclusion
si
= ci
i.e.,
◮
each player announces her production cost
◮
the lowest cost wins the market
the profit is ⌈ci ⌉c − ci
◮
◮
the second lowest cost – the lowest cost
Outline
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
Coalgebra of games
Positions
Conclusion
Equilibrium programming
Position analysis
Conclusion
Conclusion: Tasks for coalgebra of games
Coalgebra of
market games
D. Pavlovic
Games
Equilibria
◮
Positions
analyze positions as states in X
Conclusion
◮
◮
coalgebra homomorphisms between games
◮
position bisimilarity
construct equilibria as fixed points in A
s
/ A or position-wise X
◮
static 1
◮
equilibrium at a stationary position 1
s
/A
hs,xi
/ A×X