### Maximin strategy

Maximin strategy is also called low risk strategy. It is the strategy that maximizes the minimum gain that can be achieved

Example -

lets revise dominant strategy in this case

- Firm B will gain by only keeping the prices low, no matter the firm A does. So, Low price is the dominant strategy for the firm B

- Firm A doesnot have a dominant strategy. Because if it keeps the prices high, it is not gaining and if it keeps the prices low, it might any of the two.

In the above example, only firm B has a dominant strategy.Firm A doesn’t have a dominant strategy.

Maximin Strategy

- Maximin strategy maximizes the minimum gain that can be achieved.

- One firm doesn’t know what his competitor chooses.

- We know, firm 2 has a dominant strategy( low price)

- Firm A does not a dominant strategy.

- Firm A can either earn 20 or loose 100 by low prices.

- Firm A can earn 0 or loose 10 by keeping the prices high.

- Firm A can choose low prices to earn 20 but if firm B chooses high prices then firm A will loose 100.

- If firm A chooses high prices, it might earn nothing and loose 10

- The worst case scenario – firm A will loose 100.

2nd worst case – firm A will loose 10.

- Firm A can choose high prices(by loosing 10) as it will no longer have a chance of losing 100.

- So, the firm A will chosse high prices because by doing this, it is minimizing the risk that it could have faced with choosing low prices.

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Example -

| Firm B | ||

Firm A | | High price | Low price |

High price | 0.0 | -10,10 | |

Low price | -100,0 | -20,10 |

lets revise dominant strategy in this case

- Firm B will gain by only keeping the prices low, no matter the firm A does. So, Low price is the dominant strategy for the firm B

- Firm A doesnot have a dominant strategy. Because if it keeps the prices high, it is not gaining and if it keeps the prices low, it might any of the two.

In the above example, only firm B has a dominant strategy.Firm A doesn’t have a dominant strategy.

Maximin Strategy

- Maximin strategy maximizes the minimum gain that can be achieved.

- One firm doesn’t know what his competitor chooses.

- We know, firm 2 has a dominant strategy( low price)

- Firm A does not a dominant strategy.

- Firm A can either earn 20 or loose 100 by low prices.

- Firm A can earn 0 or loose 10 by keeping the prices high.

- Firm A can choose low prices to earn 20 but if firm B chooses high prices then firm A will loose 100.

- If firm A chooses high prices, it might earn nothing and loose 10

- The worst case scenario – firm A will loose 100.

2nd worst case – firm A will loose 10.

- Firm A can choose high prices(by loosing 10) as it will no longer have a chance of losing 100.

- So, the firm A will chosse high prices because by doing this, it is minimizing the risk that it could have faced with choosing low prices.

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### Nash equilibrium

To start with nash equilibrium, let us revise the dominant strategy first.

A dominant strategy is the strategy that remains the same, irrespective of the behavior of its competitor’s. In a 2 x 2 game, either both or single player have a dominant strategy.

Nash equilibrium is a situation in which a player will choose the strategy he can, knowing the competitor’s reaction and does not have any incentive to deviate.

Example

Production decision problem

| Firm B | ||

Firm A | | Produce x | Don’t produce |

Produce x | -6,-6 | 10,10 | |

Don’t produce | 10,10 | -6,-6 |

We know, that in nash eqilibriom, one company knows its competitor’s policy and does not deviate from its decision.

1. If Firm A gets to know that firm B is producing a particular commodity X.

2. Firm A will chose “not to produce” as it gives maximum profits

3. Or the Firm B gets to know that firm A is not producing the particular commodity X.

4. Firm B will chose to produce x as it gives maximum profits(10). if it also choses “not to produce” it will have losses.

5. (Don’t produce, produce) or (10,10) is a nash equilibrium situation.( both the firms will not deviate from their decision)

6. There is 1 more nash eqilibrium in the above matrix, ie (10,10)(produce ,don’t produce)

### Example: one player dominant strategy

We know that the dominant strategy is the strategy that is optimal for the firm irrespective of the behavior of the other firm.

Example of dominant strategy (when Only one player has a dominant strategy)

Steps

1. Study Firm A’s Payoff

2.Find out the maximum payoff from 2 of the strategies

3.Lets see the payoffs

(10,5) – A will earn 10, if it chooses to advertise and B also advertises.

(6.8)- – A will earn 6, if it chooses not to advertise and B advertises.

(15,0) – A will earn 15, if it chooses to advertise and B does not advertise.

(20,2) - – A will earn 20, if it chooses not to advertise and B also does not advertise.

So, here choosing advertise or not to advertise depends on B’s decision. If B decides to advertise, A will choose advertise and if B decides not to advertise, the firm A will also not advertises.

4. So, here Firm A doesn’t have a dominant strategy. Its decision depends on firm B’s decision.

5. Now, Study Firm B’s Payoff

6.Find out the maximum payoff from 2 of the strategies

7. lets study B’s payoffs

(10,5) – B will earn 5, if it chooses to advertise and A also advertises.

(6.8)- – B will earn 8, if it chooses to advertise and A does not advertise.

(15,0) – B will earn 0, if it chooses not to advertise and A advertises.

(20,2) - – B will earn 2, if it chooses not to advertise and A also does not advertise.

8.Here firm B has a dominant startegy. B is earning maximum profits when it chooses to advertise whether A advertises or not.

Example of dominant strategy (when Only one player has a dominant strategy)

| Firm B | ||

Firm A | | Advertise | Don’t Advertise |

Advertise | 10,5 | 15,0 | |

Don’t advertise | 6,8 | 20,2 |

Steps

1. Study Firm A’s Payoff

2.Find out the maximum payoff from 2 of the strategies

3.Lets see the payoffs

(10,5) – A will earn 10, if it chooses to advertise and B also advertises.

(6.8)- – A will earn 6, if it chooses not to advertise and B advertises.

(15,0) – A will earn 15, if it chooses to advertise and B does not advertise.

(20,2) - – A will earn 20, if it chooses not to advertise and B also does not advertise.

So, here choosing advertise or not to advertise depends on B’s decision. If B decides to advertise, A will choose advertise and if B decides not to advertise, the firm A will also not advertises.

4. So, here Firm A doesn’t have a dominant strategy. Its decision depends on firm B’s decision.

5. Now, Study Firm B’s Payoff

6.Find out the maximum payoff from 2 of the strategies

7. lets study B’s payoffs

(10,5) – B will earn 5, if it chooses to advertise and A also advertises.

(6.8)- – B will earn 8, if it chooses to advertise and A does not advertise.

(15,0) – B will earn 0, if it chooses not to advertise and A advertises.

(20,2) - – B will earn 2, if it chooses not to advertise and A also does not advertise.

8.Here firm B has a dominant startegy. B is earning maximum profits when it chooses to advertise whether A advertises or not.

### Example: Each player has a dominant strategy

Example of dominant strategy ( when each player has a dominant strategy)

Dominant strategy is the strategy that is optimal for the firm irrespective of the behavior of the other firm.

What is the dominant strategy for firm A and firm B?

Steps:

1. Study Firm A’s Payoff

2.Find out the maximum payoff from 2 of the strategies, no matter what the other firm does.

3. In above example, firm A earns max when it chooses to advertise.

(15,0) – A will earn 15, if it chooses to advertise and B does not advertise.

(10,5) – A will earn 10, if it chooses to advertise and B also advertises.

(6.8)– A will earn 6, if it chooses not to advertise and B advertises.

(10,2) – A will earn 10, if it chooses not to advertise and B also does not advertise.

A's maximum payoff = 15 ( and strategy = to adverise)

So, Starting a new campaign is dominant for firm A, whether firm B choses or not choses to advertise.

4. Study Firm B’s Payoff

5.Find out the maximum payoff from 2 of the strategies, no matter what the other firm does.

6. In above example, firm B earns max when it chooses to advertise.

(10,5) – B will earn 5, if it chooses to advertise and A also advertises.

(6.8)- – B will earn 8, if it chooses to advertise and A does not advertise

(15,0) – B will earn 0, if it chooses to not to advertise and A advertises.

(10,2) - – Bwill earn 2, if it chooses not to advertise and A also does not advertise.

B's maximum payoff = 8 ( and strategy = to adverise)

So, Starting a new campaign is dominant for firm B, whether firm A choses or not choses to advertise.

7. Both firms have same dominant strategies and both firms will advertise.

And this situation is called Equilibrium in dominant strategies.(ie the outcome of a game in which each firm is doing the best, regardless of what the other person is doing.)

Dominant strategy is the strategy that is optimal for the firm irrespective of the behavior of the other firm.

What is the dominant strategy for firm A and firm B?

| Firm B | ||

Firm A | | Start a new campaign | Don’t Start |

Start a new campaign | 10,5 | 15,0 | |

Don’t Start | 6,8 | 10,2 |

Steps:

1. Study Firm A’s Payoff

2.Find out the maximum payoff from 2 of the strategies, no matter what the other firm does.

3. In above example, firm A earns max when it chooses to advertise.

(15,0) – A will earn 15, if it chooses to advertise and B does not advertise.

(10,5) – A will earn 10, if it chooses to advertise and B also advertises.

(6.8)– A will earn 6, if it chooses not to advertise and B advertises.

(10,2) – A will earn 10, if it chooses not to advertise and B also does not advertise.

A's maximum payoff = 15 ( and strategy = to adverise)

So, Starting a new campaign is dominant for firm A, whether firm B choses or not choses to advertise.

4. Study Firm B’s Payoff

5.Find out the maximum payoff from 2 of the strategies, no matter what the other firm does.

6. In above example, firm B earns max when it chooses to advertise.

(10,5) – B will earn 5, if it chooses to advertise and A also advertises.

(6.8)- – B will earn 8, if it chooses to advertise and A does not advertise

(15,0) – B will earn 0, if it chooses to not to advertise and A advertises.

(10,2) - – Bwill earn 2, if it chooses not to advertise and A also does not advertise.

B's maximum payoff = 8 ( and strategy = to adverise)

So, Starting a new campaign is dominant for firm B, whether firm A choses or not choses to advertise.

7. Both firms have same dominant strategies and both firms will advertise.

And this situation is called Equilibrium in dominant strategies.(ie the outcome of a game in which each firm is doing the best, regardless of what the other person is doing.)

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