Part 1 Nocooperative Equilibria in Normal Form Games

Part 1-5 More Than Two Players

prolog• 三人賽局（ Ch 6 ）、 N 人賽局（ Ch 11 ）• Three-player games ： 二人賽局之延伸，多人賽局的起點 前者分析技巧可運用，且具備後者之發現• N-player games Many of economic issues involve considerably more than

two or three players. N-person game is more realistic. So, we need to explore games with many players.

Three-player games

• 聯盟：一群參賽者聯合協調彼此的策略稱 之為聯盟；未與其他參賽者協調合 作的參賽者則稱之為單一聯盟。• 破壞者：係指本身不可能贏，但對於誰贏 具決定性影響的參賽者。

• Coalition structure ； a partition of the players in a game into coalition, including singleton coalition （ a player acting alone ）• Two-player games Three-player game ：

coalition structure

• The possible coalition structures of this game ： i. Singleton coalition ：（ R ） ; （ S ） ;（ W ） Nash equilibrium ： ii. Grand coalition ：（ R,S,W ） How about the profile （ north, west, onshore ）？ This agreement could not be carried out (why?).

iii. Two-country coalition, another in a singleton coalition ： （R, S） ;（W）： How about the profile （（north, east） , onshore）？ Neither R or S will want to deviate their coordinated strategy and the strategy, onshore, is country W’s best response to （north, east） Similarly, the case (R,W); (S), the profile （（north, offshore） , west） is a Nash equilibrium; the case (S,W); (R), the profile （（east, offshore） , south） is a Nash equilibrium.

Remarks ：• Which of the equlibria will be observed in practice ？ If we regard the any treaty of alliance as a “clue”, every Nash equilibrium to two-country alliance is a Schelling- point Nash equilibrium.

• We would not expect to see a grand coalition in absence of enforcement mechanism because there is no Nash equilibrium to the three-country grand coalition.

• coalitions can form in noncooperative games with three or more players. However, in the absence of some enforcement mechanism, we will see only coalitions that correspond to Nash equilibria.

Be noted ！ the treaty of alliance can not deviate their individual interest, otherwise the treaty is a dead letter.

公共財捐獻賽局（ contribution games ）• Hume （ 1739 ） have understood that

if citizens respond only to private incentives,

public goods will be underprovided and public

resources overutilized. The problem of public

resources overutilized refers to tragedy of the

commons.

• dominant strategy equilibrium ： （不捐獻，不捐獻，不捐獻） 1. 三人賽局之社會困境 2. 是否存在其他聯盟之均衡解？ Eg. （（ J,K ）， L ），大聯盟？ 無執行機制時均不成立

N-players games

• The number of strategy combinations increases much more rapidly than the number of the agents. This results into very complicated games.

• we need to make some simplifying ass to make the analysis feasible.

Classification ： State variable ：• some equilibrium from the basic economics are

instances of NE with state var. and one or more type of representative agent.

the queuing game

• NE ： （ 2-sit, 4-stand ）• Is the NE （ 2-sit, 4-stand ） efficient ？ 1 the strategy profile, （ 2-sit, 4-stand ） , is a noncooperative equilibrium, but not the best solution. There exists social dilemma.

2. 忽略對他人造成之傷害， 1st-stand gets gain 5.5 ，卻使得其餘人預期報酬由 12.5 降至 11

3. 政策如何介入？即執行機制如何設計，以確 保 all-sit ？

simplifying ass. for N-person games

• Assumption 1 ： representative agent all players are identical, have the same

strategy options, and get symmetrical payoffs. note ： Even though all players have the same

strategy options and the same payoffs in a given situation, they may do different thing in equilibrium.

• Assumption 2 ： state variable. 1. it express the state of game-processing, that evolves continuously over time. 2. it sums up all information that the agent needs to choose a best-response strategy.

Games with many participants ：Proportional games

The commuters are representative agents—their payoffs vary in the same way with the number of cars—and the state variable （ congestion ） is the proportion of all commuters who drives cars.

Payoff to bus commuters ： 1-3q Payoff to car commuters ： 1.5-3.75q NE ：（ car, bus ） = （ 2/3, 1/3 ） In a NE, identical agent may choose different strategy to

maximize their payoff.