2. Lecture 4: Social welfare and symmetric
n-player games
■ Review: definition of games, strategies, NE and the technique we reach NE solution
■ Welfare maximization and Pareto dominance
■ Examples of welfare calculation
■ Symmetric n > 2 player games: Symmetric MNE
■ Examples of symmetric n-player game
■ Level-k strategy
7. Review: IEDS
■ Continue to eliminate strictly dominated strategies in this way, until you have a game
without any dominated strategies for any player.
8. Review: Mixed strategy
■ Denote ∆ 𝐴𝑖 = {𝛼𝑖}, then ∆ 𝐴𝑖 is a (|𝐴𝑖| − 1)-simplex
■ Denote ∆ 𝐴 the set of lottery over A, then ∆ 𝐴 is a (|A| − 1)-simplex
■ Then, each prole of mixed strategies 𝛼 = (𝛼1, … , 𝛼 𝑛), maps to a lottery on ∆ 𝐴 ,
with assumption of independence, we have
11. Welfare notations
■ Setting 𝜌𝑖= 𝜌𝑗 for each 𝑖, 𝑗 ∈ 𝑁 gives the “utilitarian solution".
■ Fact 1: Welfare-maximizing outcomes are Pareto-efficient.
■ Fact 2: Takashi Negishi and Hal Varian proved the converse: For Pareto-efficient a,
there exist (𝜌𝑖)𝑖∈𝑁 s.t. a is welfare maximizing.
12. Expected welfare notions for mixed extensions
■ Outcome a is “Pareto efficient" if it is not weak Pareto dominated.
■ Setting 𝜌𝑖= 𝜌𝑗 for each 𝑖, 𝑗 ∈ 𝑁 gives the “utilitarian solution".
■ Facts 1 and 2 continue to hold.
13. Welfare in “Battle of the Sexes"
■ Welfare (𝑈𝑖
∗
𝑝∗, 𝑞∗ , 𝑈𝑗(𝑝∗, 𝑞∗)) in Nash equilibrium (𝑝∗, 𝑞∗):
■ PNE (1, 1): (2, 1).
■ PNE (0, 0): (1, 2).
■ MNE (2/3, 1/3):
■ Large welfare losses to miscoordination!
14. Symmetric n > 2 player games: Symmetric
MNE
■ The symmetric MNE could be a PNE.
15. Symmetric n > 2 player games: Stag Hunt
■ n > 2 hunter-gatherers (players) coordinate to feed themselves.
■ Each player can either participate in a (S)tag hunt, or stay home and (G)ather food:
𝐴𝑖 = 𝑆, 𝐺 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖 ∈ 𝑁
■ A successful stag hunt, which requires M > 1 or more participants (𝑛 ≤ 𝑀), yields
total value V > 2.
■ Denote # of hunters #S = # (𝑎𝑖 = 𝑆, 𝑖 ∈ 𝑁)
■ Payoffs to 𝑖 ∈ 𝑁 are given by:
■ PNE (1): 𝑎𝑖
∗
= 𝐺 for each 𝑖 ∈ 𝑁
■ This PNE is Pareto inefficient. Consider when there are exactly two people
participate stag hunt…
16. Symmetric n > 2 player games: Stag Hunt
■ PNE (2):
■ Case 1) V > n: 𝑎𝑖
∗
= 𝑆 for each 𝑖 ∈ 𝑁
■ Case 2) V < n: : 𝑎𝑖
∗
= 𝑆 for exactly m players where
𝑉
𝑚
≥ 1and V /(m + 1) < 1.
■ This gives (
𝑛
𝑚∗) (i.e. n choose m) different PNE: there are (
𝑛
𝑚∗) ways to decide on
which m players hunt the stag.
■ Provided 𝑉 ∉ ℤ, hunters realize value above 1: Pareto efficient.
19. Cournot Duopoly
■ Two firms i and j, compete by producing quantities 𝑞𝑖 ≥ 0, and 𝑞 𝑗 ≥ 0. under price
function 𝑃 𝑞𝑖, 𝑞 𝑗 = 𝑎 − 𝑏(𝑞𝑖 + 𝑞 𝑗), 𝑎, 𝑏 > 0. Marginal cost of production c>0
■ Each firm maximizes profit given the other's production. For firm i :
■ First-order condition: 𝑎 − 2𝑏𝑞𝑖 − 𝑏𝑞 𝑗 − 𝑐 = 0
■ Solving for 𝑞𝑖 gives firm i's best response:
■ Likewise, we can derive firm j’s best response
24. Cournot Duopoly
■ ...IESDS converges on the unique PNE in the Cournot duopoly.
■ Consequently, no additional MNE exist (i.e. no MNE with mixing).
25. Level-k strategy
■ Formally, Level-k strategies are given by the multilateral best-response dynamic
starting from v.
■ Level-k strategy gives optimal play when all other players play according to level-(k-1)
strategy.
■ Players are “bounded rational" because they believe that others believe that others
believe...that players play a random strategy.
26. Summary
■ Welfare reflects the benefits at the aggregate level. We consider the aggregate utility
because we want to understand the outcomes of social miscoordination
■ Pareto dominance and welfare maximization
■ NE in symmetric games: easy to solve because everyone’s strategy is going to be the
same
■ Examples: battle of sexes, Cournot duopoly
■ Level-k strategy
