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Game Theory

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Game Theory

Part 4: Dominant Strategies

- Sometimes in a matrix game, a player will have a strategy that, given all of the resulting outcomes, would not be worth playing.
- Such a strategy would not be worth playing if it is never better and sometimes worse than some other strategy, regardless of the strategies of other players.
- For a given player, strategies that are never better and sometimes worse than other strategies are called dominated strategies. (We can think of this as equal or worse than all of the other strategies.)
- On the other hand, a dominant strategy is one that is sometimes better and never worse than all other strategies, regardless of the strategies of the other players. (We can think of this as equal or better than all of the other strategies.)

player 2

- Does either player have a dominant strategy in the game below?

For player 1, Z is a dominant strategy.

This is because strategy Z is always better than either of the other two strategies.

Because Z is dominant, the other strategies must be dominated.

We can also say that, for player 1, strategy X dominates strategy Y.

player 1

For player 2, strategy A is dominant because A is never worse and, in this case, always better than strategy B from the standpoint of minimizing payoff values.

Because A is dominant, then B is dominated.

player 2

For player 1, there is no dominant strategy.

However, Z is a dominated strategy.

This is because, while no strategy dominates all the others, for player 1, X and Y are never worse and sometimes better than Z.

- Does either player have a dominant strategy in the game below?

player 1

For player 2, there is also no dominant strategy.

There is also no dominated strategy for player 2 because there is no strategy that is never better and sometimes worse than any other strategy.

(Note that because there are only two strategies for player 2, knowing that there is no dominant strategy, we know there is also no dominated strategy.)

- If a player has two strategies,
- if one strategy is dominant, then the other is dominated.
- if neither is dominant, then neither is dominated.

- If a player has three or more strategies,
- if one strategy is dominant, then all others are dominated.
- If one strategy is dominated, there may or may not be dominant strategies.

- Dominant – better than or equal to any other strategy.
- Dominated – worse than or equal to any other strategy.

player 2

Does either player have a dominant strategy for this game ?

Player 1 has no dominant and no dominated strategy.

Why?

player

1

- Dominant – better than or equal to any other strategy.
- Dominated – worse than or equal to any other strategy.

player 2

Player 1 has no dominant and no dominated strategy…If player 2 chose A, then,

for player 1,

X is the worst strategy

Y is the best strategy

Z is neither better or worse than any other

player

1

- Dominant – better than or equal to any other strategy.
- Dominated – worse than or equal to any other strategy.

player 2

Player 1 has no dominant and no dominated strategy…If player 2 chose B, then,

for player 1,

X is neither better or worse than the others

Y is the worst strategy

Z is the best strategy

player

1

- Dominant – better than or equal to any other strategy.
- Dominated – worse than or equal to any other strategy.

player 2

Player 1 has no dominant and no dominated strategy…If player 2 chose C, then,

for player 1,

X is neither better or worse than any other

Y is the best strategy

Z is the worst strategy

player

1

- Dominant – better than or equal to any other strategy.
- Dominated – worse than or equal to any other strategy.

player 2

Player 1 has no dominant and no dominated strategy.

This is because there is no strategy that is equal or better than the others and no strategy that is equal or worse than the others.

player

1

player 2

Now, lets consider player 2…

Does player 2 have a dominant or any dominated strategies ?

player

1

player 2

Now, lets consider player 2…

If player 1 chose strategy X,

then for player 2 (who wants to minimize values)

A is best and C is worst

player

1

player 2

Now, lets consider player 2…

If player 1 chose strategy Y,

then for player 2 (who wants to minimize values)

B is best and C is worst

player

1

player 2

Now, lets consider player 2…

If player 1 chose strategy Z,

then for player 2 (who wants to minimize values)

C is best and B is worst

player

1

player 2

For player 2, there is no dominant and no dominated strategy.

This is because looking at potential payoffs for each of the strategies, A, B or C, …

there is no strategy that is never worse and sometimes better

and no strategy that is never better and sometimes worse

player

1

- Recall the example zero-sum game that modeled the Battle of the Bismark Sea. In strategic form, the game is as follows:

For the Japanese Admiral, there is a dominant strategy.

Remember, the Japanese Admiral, being the column player, wants to minimize the payoff value.

With these two strategies, for the Japanese Admiral, traveling north is the dominant strategy because it is never worse and sometimes better than traveling south.

Japanese Admiral

American General

For the American General, there is no dominant strategy: For the American’s, the best strategy will depend on the strategy of the Japanese.

- It is interesting to see how the dominant strategy for the Japanese would be identified visually if we attempted to find a mixed strategy equilibrium point.

As before, we could assign probabilities to each strategy and find expected payoff functions for each strategy for each player.

For the American’s, we’d have the following payoff functions for the Japanese

EN = 2q + 1(1-q) and

ES = 2q + 3(1-q)

Japanese Admiral

p

1-p

q

American General

1-q

For the Japanese we’d get the following

EN = 2p + 2(1-p) and ES = 1p + 3(1-p)

Recall – each represent payoff values to the opposing player.

- Graphing payoff functions for each player, we can identify which has a dominant strategy and which does not. To graph, we’ll simplify each function first…

For the American’s we have:

EN = 2q + 1(1-q)

= 2q + 1 – 1q

= q + 1

and

ES = 2q + 3(1-q)

= 2q + 3 – 3q

= -q + 3

For the Japanese we have:

EN = 2p + 2(1-p)

= 2p + 2 – 2p

= 2

and

ES = 1p + 3(1-p)

= 1p + 3 – 3p

= -2p + 3

- Graphing payoff functions for each player, we can identify which has a dominant strategy and which does not.

For the American’s we have:

EN = q + 1 and ES = -q + 3

For the Japanese we have:

EN = 2 and ES = -2p + 3

ES

3

3

EN

2

2

EN

1

ES

1

p

q

1

1

no dominant strategy for Americans:

highest payoff depends on p, a choice of the Japanese

These lines represent payoffs to the Japanese – clearly EN is dominant because it is lower.

- Consider the following matrix game:

financial market

investor

financial market

investor

Suppose payoffs represent thousands of dollars gained or lost on a particular day.

In this 3x3 matrix game, the investor has a dominant strategy: The mixed portfolio strategy is dominant because it is never worse and sometimes better than any other strategy, regardless of the choice of the financial market.

In this case, the financial market (column player) has no dominant strategy. However, in this example, “mixed portfolio” and “no change in market” form an equilibrium point.

Company B

Company A

Suppose two companies face each other in a competitive market. Suppose each has three strategies, as shown above, and the payoffs, representing profits in hundreds of thousands of dollars for a given year, are as given in the table.

Notice that, in this example, both companies have dominant strategies of raising prices. Because both players have dominant strategies, these form the equilibrium point of the game.

A game in which potential payoffs to each player is the same is a symmetric game. Because potential payoffs are different in this game, it is an example of a nonsymmetrical game.

Company B

Company A

This is an example of a symmetric game. Potential payoffs to each player are the same.

Both players have dominant strategies in lowering prices. So the combination of both strategies is an equilibrium point.