Coalition Formation and Price of Anarchy in CournotOligopoliesNicole Immorlica, Evangelos Markakis, and Georgios Piliouras

Presentation By Sur Samtani

Cournot CompetitionAntoine Augustin Cournot (1801-1877) studies competition in local spring water duopoly

Each firm’s production affects market price

Outline1.Introduce Traditional Cournot Model Formally2.Determine “Price of Anarchy”3.Introduce Modified Cournot Model with Coalition Formation4. Determine “New Price of Anarchy”

1 2 n

Traditional Cournot CompetitionFirms

. . .

Strategy profile q= (q1, q2, …, qn) Strategy Space: R+

Market

q1 q2 qn

C C C

P(q) = max{0, a-b(q1+q2+…+qn)}

ui = pqi-cqi

c = per unit cost

Relevant TermsSocial Welfare: Total Profit of All Firms Nash Equilibrium: A strategy profile (q1, q2, …, qn) for which no firm i has an incentive to deviate from producing qi (profit will not increase by deviating)Price of Anarchy:

Cournot Model ContinuedFor symmetric cost c and linear price function:

}(a,b,c ∊ R and a>c) P

𝑝 (𝑞 )

c

Qqm

pm

q*

p*

Price of Anarchy in CournotSocial Welfare of Socially Optimum = = E• ∀i, qi = q*• ∀i, ui = Social Welfare of Unique N.E. = = F• Price of Anarchy (P.O.A.) = = (linear with respect to the number of firms)

Coalition Formation Model

. . .

S1 S2 Sk

= Firm

= Coalition

𝑢𝑖=𝑢𝑆 𝑗

¿𝑆 𝑗∨¿¿

for firm i in coalition Sj

Partition of Coalitions ∏= (S1, S2, …, Sk)

Legal Moves: Type 1Si Si

’

Successful move if and only if all firms forming coalition Si

’ increase their payoffs

Legal Moves: Type 2Si Sj

Successful if and only if all deviating members of Si and all members of Sj improve payoff

Legal Moves: Type 3

Successful if and only if all members of merging coalitions increase payoff

Successful Deviations (cont.)A partition ∏ is stable if there exists no successful deviation of any type

New Coalitional Price of Anarchy:

Restricted Moves• Type 1: • Type 2: • Type 3:

A member of a coalition of ∏, decides to form a singleton coalition on his own. The coalition from which the player left dissolves into singleton players. A member of a coalition of ∏ decides to leave its current coalition Si (where |Si | ≥ 2), and join another coalition of ∏, say Sj . The rest of coalition Si dissolves into singleton players. A set of singleton players of ∏ decide to unite and form a coalition.

Roadmap for Rest of Talk1. Find Conditions on stability for each type of move2. Construct an optimization problem to maximize k

(number of coalitions) given that these conditions hold

3. This will give us the value of k for the worst stable outcome, and we will find the Coalitional Price of Anarchy

Establishing Conditions for StabilityLet si =|Si| for Si∊∏, and assume k ≥ 2 and si ≥ 2,For a player j ∊ Si, j can make a Type 1 move (form his own coalition) or Type 2 move (join existing coalition)• Payoff of Type 1 move for player j : • Payoff of Type 2 move for player j : Since k ≥ 2 and si ≥ 2, u ≥ u’Type 1 move is always the most profitable deviation

Establishing Conditions for Stability (Cont.)

•Current payoff of player in coalition Si: •Payoff of most profitable deviation: •For stability, we must have v≥u , ∀i

Establishing Conditions for Stability (Cont.)Suppose that we have k1 singleton coalitions and k2 non-singleton coalitions:

• Current payoff for singleton coalition: • Payoff after Type 3 move (merge with other singleton coalitions): • Current partition is stable if and only if ≥ ≥

Summary of Stability Constraints

, ∀i≥

k coalitions

k1 singleton coalitions

k2 non-singleton coalitions

si = |Si|

n = number of firms

Optimization Problem

s.t.

≥

k coalitions

k1 singleton coalitions

k2 non-singleton coalitions

si = |Si|

n = number of firms

Price of AnarchyLP yields k1 ≤ and k2 ≤ k ≤ + •Upper Bound for P.O.A.= • Separately Calculated Lower Bound for P.O.A.= •New model results in lower Price of Anarchy than original standard Cournot Model with no Coalition Formation•Conclusion: Firms’ flexibility to collude results in lower P.O.A

Further Research•More natural coalition formation models (coalitions

don’t break up into singletons)

•Price of Anarchy with non-linear price function

•Price of Anarchy with different model for competition between firms (Bertrand Price Competition)

End