# Lec5 最近の更新履歴 yyasuda's website

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## Review of Lecture 4 

  Spatial competition model has a unique NE in which both shops choose the middle point.

Minimal differentiation!

  Political platforms (left/right or liberal/ conservative)

  TV shows (time slots)

  Under the Bertrand model:

  Price will be set at the level of marginal cost even when there are only two competitors.

Huge jump from 1 (monopoly) to 2 (duopoly)!

  Under the Cournot model:

  Total output (price) is between monopoly level and that under perfect competition.

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## Zero-Sum Game 

  Matching Pennies:

Player 2 Player 1

-1 1

Tails -1 1

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## No Nash Equilibrium?

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### Examples are:

  Porker: bluff or not

  Battle (War): attack/defend by land or by sea

  Tennis: left or right to serve/receive

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## Matching Pennies Again 

  Introducing mixed strategies:

Player 2 Player 1

1 -1

-1 1

Tails (1-q)

-1 1

1 -1

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## How to Find Equilibrium?

  If a player takes both “Heads” and “Tails” with positive probability, she must be indifferent between these two pure strategies, i.e., the expected payoff derived by

choosing Heads must be equal to that by choosing Tails.

  Otherwise, it is strictly better for her to choose either H or T for sure; no incentive to mix the strategies.

  -p + (1-p) = p - (1-p), hence p = 0.5.

  q - (1-q) = -q + (1-q), hence q = 0.5.

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## Best Response Curves 

  Given the opponent mixed strategy, draw the best response (mixed strategy) against it.

p

1

0.5

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## Existence of NE

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### A: Yes. For example, integer game.

  (Two person) Zero-sum game in which players independently choose a number simultaneously. The player who picked a larger number wins.

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## Modified Matching Pennies 

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### following:

Player 2 Player 1

2 -2

-1 1

Tails -1 1

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## Indifference Property

  Under mixed-strategy NE, Player 1 must be indifferent between choosing H and T:

-2p’ + (1-p’) = p’ - (1-p’), hence p’ = 0.4.

  Similarly, Player 2 must be indifferent between choosing H and T:

2q’ - (1-q’) = -q’ + (1-q’), hence q’ = 0.4.

  You can easily verify that (p’, q’) = (0.4, 0.4) indeed becomes a mixed-strategy NE.

  Note that player 2 chooses H with less than a half chance, even if H looks “better” strategy than T.

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## Multiple Nash Equilibria 

2 1

D (x) E (y) F (1-x-y)

A (a)

6 7

8 5

0 0

B (b)

8 5

6 7

0 0

C 0 0 4

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## Three Different Equilibria 

  Pure-strategy NE

  (C, F) is a unique pure-strategy NE.

  Mixed-strategy NE (1)

  P1 mixes A (q) and B (1-q); p2 mixes D (p) and E (1-p).

  7p + 5(1-p) = 5p + 7(1-p) p = 1/2.

  6q + 8(1-q) = 8q + 6(1-q) q = 1/2.

  Mixed-strategy NE (2)

  Both players use all three strategies.

  A (a), B (b), C (1-a-b); D (x), E (y), F (1-x-y)

  7x + 5y = 5x + 7y = 4(1-x-y) x = y = 1/5.

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## Example in Business Competition (1) 

  R&D Race Game

  2 firms simultaneously decide the amount of investment

  Possible strategies are 0, 1, 2, (million dollars).

  The larger investment => Winner; receive 3 (as a revenue)

  The smaller investment => Loser; receive 0

  In the case of tie => both receive 1

  Is there any pure strategy NE?

  Derive mixed strategy NE.

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## Example in Business Competition (2) 

  Simultaneous Entry Game

  3 firms simultaneously choose entry decision.

  Possible strategies are enter (E) or not (N).

  If only 1 firm enter => profit is 4

  If 2 firms enter => profit is 1 each

  If all 3 firms enter => profit is -2 each

  Choosing N results in 0 profit.

  Is there any pure strategy NE?

  Are there multiple mixed strategy NE?

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## Example: Soccer (Penalty Kick) 

Kicker GK

Left (x) Middle (y) Right (1-x-y)

Left (p)

40 60

100 0

80 20

Middle (q)

80 20

0 100

80 20

Right 80 100 20

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## Further Exercises 

  Construct a game which has both pure-strategy and mixed-strategy Nash equilibria.

  Consider a modified matching penny game in which the payoff on (H, H) is changed from (-1, 1) to (-k, k) for k > 0. Then, derive a mixed-strategy Nash equilibrium.

  Prove that a strategy which is erased by the iterated

elimination of strictly dominated strategies can never be selected (with positive probability) in a mixed-strategy Nash equilibrium.

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## 参照

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