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ECO290E: Game Theory

Lecture 3: Why and How is Nash Equilibrium Reached?

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Review of Lecture 2

  Each (normal-form) game consists of 1) players, 2) strategies, and 3) payoffs.

  Any two players game with finite strategies can be expressed by a payoff (bi-) matrix.

  In a Nash equilibrium, no one has an incentive to change her strategy unilaterally.

  A Nash equilibrium is not necessary Pareto efficient.

  Ex) Prisoner’s Dilemma

  There can be multiple Nash equilibria.

  Ex) Coordination game

  One equilibrium can be less efficient than (Pareto dominated

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Four Reasons for NE

There are at least four reasons why we can expect Nash equilibrium (NE) would realize.

1. By rational reasoning

2. Being an outstanding choice

3. A result of discussion

4. A limit of some adjustment process

Which factor serves as a main reason to achieve NE depends on situations.

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1. Rationality

  Players can reach Nash equilibrium only by rational reasoning in some games, e.g., Prisoners’ dilemma.

  However, rationality alone is often insufficient to lead to NE. (see Battle of the sexes, Chicken game, etc.)

  A correct belief about players’ future strategies

combined with rationality is enough to achieve NE.

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2. Focal Point

  A correct belief may be shared by players only from individual guess.

  Class room experiments:

  Choose a subway station in Tokyo and write down its name.

  You will win if you can choose the most popular answer.

  Most of the students are expected to write “xxx”.

  Like this experiment, there may exist a Nash equilibrium which stands out from the other equilibria by some

reason.

Focal Point (by Thomas Schelling)

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3. Self-Enforcing Agreement

  Without any prize or punishment, verbal promise achieves NE while non equilibrium play cannot be enforced.

NE = Self-Enforcing Agreement

Example: Prisoner’s Dilemma

  Even if both players promise to choose “Silent,” it will not be enforced since (S,S) is not a NE.

  (C,C) is self-enforced (although there is no point of making such a verbal promise…).

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4. Repeated Play

  Through repeated play of games, experience can generate a common belief among players.

Example 1: Escalator

  Either standing “right” or “left” can be a NE.

Example 2: Keyboard

  “Qwerty” vs. “Dvorak”

History of adjustment processes determine which equilibrium is realized: Economic history has an

important role.

“Path Dependence” (by Paul David)

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Roles of Social Science

  Explain the reason behind the frequently observed phenomena.

NE serves as a powerful tool.

  Predict what will happen in the future.

Although it is usually difficult to make a one-shot

prediction, NE may succeed to predict the stable situation after some (long) history of adjustment processes.

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What is Rationality?

 

A player is rational if she chooses the strategy

which maximizes her payoff (given other players’

strategies.)

  Implicitly assume that we can describe her preferences as payoff numbers.

The definition implies that a rational player

 

takes a dominant strategy whenever it is available.

 

never takes (strictly) dominated strategies.

  If strategy s generates always higher payoff than s’, then s’ is called (strictly) dominated by s.

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Solving Game by Rationality

 Player 2 Player 1

Left Middle Right

Up 0

1

2 1

1 0

Down 3 1 0

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Iterated Elimination Process

  What can players deduce if they completely understand (i.e., agree) that they are both rational players?

  Neither of them will take strictly dominated strategies…

  Step 1: Right is strictly dominated by Middle, so player 2 NEVER takes Right.

  Step 2: Given the belief that Player 2 never takes Right, Down is strictly dominated by Up. Therefore, Player 1 will NOT take

Down.

  Step 3: Given the belief that Player 1 will not take Down, Left is strictly dominated by Middle. Therefore, Player 1 will NOT take Left.

  Step 4: Only (Up, Middle) is survived after the iterated elimination process!

This reasonable solution coincides with NE.

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Limitation of Rationality

  The process often produces a very imprecise prediction about the play of the game.

  Coordination, Battle of the sexes, Chicken, etc.

  Nash equilibrium is a stronger solution concept than iterated elimination of strictly dominated strategies, in the sense that the players’ strategies in a Nash

equilibrium ALWAYS survive during the process, but the converse is not true.

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Is it Solved by Rationality?

 Player 2 Player 1

Left Middle Right

Up 0

1

1 1

3 1

Down 3 0

1 2

0 0

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Randomized Strategies 

  No strategy looks to be dominated…

  If a player 2 randomizes L and R with 50% each, then

  Such mixed (randomized) strategy yields 1.5 (as an expected payoff) while M gives 1 irrespective of player 1’s strategy.

  Therefore, M is eliminated by mixing L and R.

  After eliminating M, we can further eliminate D (step 2) and L (step 3), eventually picks up (U, R) as a unique outcome.

  We essentially extend the concept of strategy, i.e.,

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Spatial Competition Model

  Players: Two ice cream shops

  Strategies: Shop location along a beach (any integer between 0 and 100)

  Payoffs: Profits = The number of customers

Assumptions:

  Customers are located uniformly on the beach.

  Each customer goes to the nearest shop (and buys exactly one ice dream).

  If both shops choose the same location, each receives half of the customers.

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Nash Equilibrium

  There is a unique NE in which both shops open at the middle, i.e., s1 = s2 = 50.

  Called “Principle of minimum differentiation”.

Why?

1. Choosing separate locations never becomes a NE.

2. Choosing the same locations other than the middle point also fails to be a NE.

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Applications of Spatial Competition 

  Spatial competition (Hotelling) Model can explain

  Two players competing in the same platform

  Aiming for as much customers each could obtain

  Results in taking very similar strategy one another…

Players Strategies Phenomena

Two Big Political Parties Target Policy Moderate Politics

Convenience Stores Location Two Stores Locate Nearby

TV Programs Broadcasting Time Same Topics on same time for Different Channels

Manufacturers Taste or Appearance Similar Products (Coke, Audio Players, Smartphone, etc)

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Is There a Dominant Strategy? 

  Q: Is 50 a dominant strategy?

  A: No!

  If 50 is a dominant strategy, then payoff of choosing 50 must be strictly larger than the payoff of choosing ANY other

strategy, for EVERY possible strategies of the other player.

  Suppose other player’s strategy is 90. Then, choosing 50 yields payoff of 70 (= (50+90)/2), while choosing 80 will achieve 85 (= (80+90)/2)

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Solution by Iterated Elimination

  Step 1: A rational player never takes the edges, since 0 (100) is strictly dominated by 1 (99).

  Step 2: 1 and 99 are never chosen if the players know their rival is rational.

  Step 3: 2 and 98 are never chosen if the players know that their rival knows that you are rational.

  Step 50: 49 and 51 are never chosen if the players know that their rival knows that …

Both players choose 50 in the end!

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Further Exercises 

  Find a real life example that can be described as a game with multiple Nash equilibria in which focal point story picks up a unique equilibrium.

  List up real life stories of path dependence.

  Consult an advanced textbook and study the idea of

“rationalizability” (each player only chooses a strategy that can be a best response to opponents’ possible

strategies) and its connection to the iterated elimination of strictly dominated strategies.

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