Game theory can be defined as "a way to study human behavior through mathematics, economics, and other social and behavioral sciences" (Stewart). It tries to determine, through the use of mathematics and logic, actions that players should take in order to receive the best possible outcomes for themselves. If the best choice remains constant regardless of others' actions, a dominant strategy exists (Dixit). A dominant strategy is the course of action "that results in the highest payoff to a player regardless of the opponent's action" (Baye, 355). The example below shows a dominant strategy.

If student 1 goes first and chooses to study, student 2 will also choose to study. This gives student 2 an A rather than the D they would have received if they did not study. If student 1 chooses to not study, student 2 will choose to study so they get a B instead of an F. Student 2 obviously has a dominant strategy - to study. If this game is played again with student 2 leading, student 1 would also have the dominant strategy of studying. This dominant strategy also marks this game's Nash equilibrium, or the "set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other players' strategies" (Baye, 357).

Types of Dominant Strategies

Strictly Dominant

A strictly dominant strategy is one where "regardless of what any other players do, the strategy earns a player a strictly higher payoff than any other" (Shor). However "rational players do not play strictly dominated strategies and so, once you determine a strategy is strictly dominated by other strategy, simply remove it from the game" (Strictly). This process is known as iterated elimination of dominated strategies. This will be discussed more fully later.

Example

The most well-known example of a strictly dominant strategy game is that of the Prisoner's Dilemma. This is a classic game where "two players may each "cooperate" with or "defect" (i.e. betray) the other player" (Prisoner's). Although each player would receive a better payoff if one confesses and the other does not, this payoff is inferior to neither confessing. Thus, each player is better off if they cooperate and serve a short-term sentence.

Weakly Dominant

A weakly dominant strategy is one where "no matter what the rival does, there is some strategy that does equally well and sometimes strictly better" (Dominance). A weakly dominant strategy can sometimes be rational to use if you know your opponents' strategy.

Example

The above payoff matrix gives an example of a weakly dominant strategy. For player 2, playing game B is weakly dominated by game A. However, if player 2 believes that player 1 will choose bottom, the choice of game B by player 2 is a rational choice.

Iterated Elimination of Dominated Strategies

IEDS is one common technique that can be used to solve a game. This technique involves the removal of dominant strategies from a game to simplify a game. This process can identify the Nash Equilibrium(s). There are two versions of IEDS: one that eliminates only strictly dominant strategies and one that eliminates all (strictly and weakly) dominant strategies.

The version that removes only strictly dominated strategies is used most frequently. By removing these strategies, the game is reduced down to the Nash equilibrium and thus gives the best possible payoff decision that the players can choose for everyone (Dominance (Game Theory)).

The version that removes all dominant strategies faces more risk. The "elimination of weakly dominated strategies may eliminate some Nash equilibrium(s). As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium" (Dominance (Game Theory)).

Dominant Strategy vs Nash Equilibrium

Any dominant strategy is a Nash Equilibrium. However, a Nash Equilibrium is not defined as a player's dominant strategy. An example of a Nash Equilibrium not being a dominant strategy is presented in the previous section. When applying IEDS, the dominant strategies - either strict or weak - are eliminated in order to identify the Nash Equilibrium(s). Therefore, not all Nash Equilibriums are dominant strategies. The picture below shows the relationship between the dominant strategy, IEDS, and the Nash Equilibrium.

(Solving)

If there is a dominant strategy solution then that solution is also an IEDS solution and a Nash equilibrium.

If there is an IEDS solution, then it is also a Nash equilibrium. Note that dominant strategy solutions are a proper subset of IEDS solutions. Even when there is no dominant strategy solution, there may be an IEDS solution.

Even when there are no dominant strategy or IEDS solutions, there may be a Nash equilibrium. (Solving)

No Dominant Strategy?

If a game is lacking any dominant strategies, there exists at least one player who would wish to change his or her own strategy based on another players' strategy. In the absence of a dominant strategy, a player will still want to play in the way that gives him or her the best possible outcome.

Importance of Dominant Strategies

If a player knows their opponents' payoff matrix, the existence of a dominant strategy can be determined. For example, if the conclusion can be made that a player will always (dominantly) choose the same action, the choice that another player will make can be modified to maximize his or her own payoff. The payoff matrix below depicts this strategy.

In a sequential move game, if player 1 (follower) has the dominant strategy to always choose A, player 2 (leader) can choose the action that results in the better payoff for themself - in this case B. If the knowledge of player 1's strategy were not known to player 2, player 2 may not have chosen the best possible action for themselves. Having the knowledge of the other players' strategy makes all the difference with your own strategy. Keep in mind that this knowledge can also be used against you if you are the one with the dominant strategy.

Multiple Choice Questions

1. If a Nash Equilibrium exists which of the following is (are) true?

a. there are at least three players

b. there is no better payoff given others' actions

c. the game is imaginary

d. none of the above

e. all of the above

The correct answer is b. A Nash Equilibrium is the best a player can do given another player's strategy.

2. What type of strategy do rational players not choose to play?

a. below dominant

b. weakly dominant

c. unplayable dominant

d. strictly dominant

e. absolutely dominant

The correct answer is d. A strictly dominant strategy is one that rational players do not play.

3. In a two player game, the question of whether to advertise or not is asked. If both players advertise, they each earn a $5 payoff. If one advertises and the other does not, the one that advertises earns a $10 payoff and the other receives a $0 payoff. If neither advertises, they each earn a $3 payoff. Which of the following is (are) true?

a. player 1 has a dominant strategy

b. neither player has a dominant strategy

c. player 2 has a dominant strategy

d. there is a dominant strategy to not advertise

e. both a and c

The correct answer is e. Both players have a dominant strategy to advertise.

4. What term in game theory describes an action that remains constant regardless of an opponent's action?

a. fixed

b. dominant

c. incorrect

d. normal

e. none of the above.

The correct answer is b. A dominant strategy is one where the best choice for a player is constant no matter the action another player takes.

5. Two players earn $10 if they each do action A, $15 if they each do action B, $6 if one does action A and the other action B, and $3 if one does action B and the other action A. If player 1 chooses action B, what is the best possible choice for player 2?

a. Action A

b. Action B

c. both a and b

d. not enough information to tell

e. no action

The correct answer is b. If player 1 chooses action B, player 2 can earn $6 or $15. Player 2 should choose to earn $15 by doing action B.

References

Baye, Michael R., "Managerial Economics and Business Strategy", McGraw-Hill Irwin. 2006.
Dixit, Avinash and Barry Nalebuff., "Game Theory".

Dominant StrategyGame theory can be defined as "a way to study human behavior through mathematics, economics, and other social and behavioral sciences" (Stewart). It tries to determine, through the use of mathematics and logic, actions that players should take in order to receive the best possible outcomes for themselves. If the best choice remains constant regardless of others' actions, a dominant strategy exists (Dixit). A dominant strategy is the course of action "that results in the highest payoff to a player regardless of the opponent's action" (Baye, 355). The example below shows a dominant strategy.

If student 1 goes first and chooses to study, student 2 will also choose to study. This gives student 2 an A rather than the D they would have received if they did not study. If student 1 chooses to not study, student 2 will choose to study so they get a B instead of an F. Student 2 obviously has a dominant strategy - to study. If this game is played again with student 2 leading, student 1 would also have the dominant strategy of studying. This dominant strategy also marks this game's Nash equilibrium, or the "set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other players' strategies" (Baye, 357).

Types of Dominant Strategies

A strictly dominant strategy is one where "regardless of what any other players do, the strategy earns a player a strictly higher payoff than any other" (Shor). However "rational players do not play strictly dominated strategies and so, once you determine a strategy is strictly dominated by other strategy, simply remove it from the game" (Strictly). This process is known as iterated elimination of dominated strategies. This will be discussed more fully later.Strictly Dominant## Example

The most well-known example of a strictly dominant strategy game is that of the Prisoner's Dilemma. This is a classic game where "two players may each "cooperate" with or "defect" (i.e. betray) the other player" (Prisoner's). Although each player would receive a better payoff if one confesses and the other does not, this payoff is inferior to neither confessing. Thus, each player is better off if they cooperate and serve a short-term sentence.

A weakly dominant strategy is one where "no matter what the rival does, there is some strategy that does equally well and sometimes strictly better" (Dominance). A weakly dominant strategy can sometimes be rational to use if you know your opponents' strategy.Weakly Dominant## Example

The above payoff matrix gives an example of a weakly dominant strategy. For player 2, playing game B is weakly dominated by game A. However, if player 2 believes that player 1 will choose bottom, the choice of game B by player 2 is a rational choice.

IEDS is one common technique that can be used to solve a game. This technique involves the removal of dominant strategies from a game to simplify a game. This process can identify the Nash Equilibrium(s). There are two versions of IEDS: one that eliminates only strictly dominant strategies and one that eliminates all (strictly and weakly) dominant strategies.Iterated Elimination of Dominated StrategiesThe version that removes only strictly dominated strategies is used most frequently. By removing these strategies, the game is reduced down to the Nash equilibrium and thus gives the best possible payoff decision that the players can choose for everyone (Dominance (Game Theory)).

The version that removes all dominant strategies faces more risk. The "elimination of weakly dominated strategies may eliminate some Nash equilibrium(s). As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium" (Dominance (Game Theory)).

Any dominant strategy is a Nash Equilibrium. However, a Nash Equilibrium is not defined as a player's dominant strategy. An example of a Nash Equilibrium not being a dominant strategy is presented in the previous section. When applying IEDS, the dominant strategies - either strict or weak - are eliminated in order to identify the Nash Equilibrium(s). Therefore, not all Nash Equilibriums are dominant strategies. The picture below shows the relationship between the dominant strategy, IEDS, and the Nash Equilibrium.Dominant StrategyvsNash Equilibrium

If a game is lacking any dominant strategies, there exists at least one player who would wish to change his or her own strategy based on another players' strategy. In the absence of a dominant strategy, a player will still want to play in the way that gives him or her the best possible outcome.No Dominant Strategy?

If a player knows their opponents' payoff matrix, the existence of a dominant strategy can be determined. For example, if the conclusion can be made that a player will always (dominantly) choose the same action, the choice that another player will make can be modified to maximize his or her own payoff. The payoff matrix below depicts this strategy.Importance of Dominant Strategies

In a sequential move game, if player 1 (follower) has the dominant strategy to always choose A, player 2 (leader) can choose the action that results in the better payoff for themself - in this case B. If the knowledge of player 1's strategy were not known to player 2, player 2 may not have chosen the best possible action for themselves. Having the knowledge of the other players' strategy makes all the difference with your own strategy. Keep in mind that this knowledge can also be used against you if you are the one with the dominant strategy.Multiple Choice Questions1. If a Nash Equilibrium exists which of the following is (are) true?

- a. there are at least three players
- b. there is no better payoff given others' actions
- c. the game is imaginary
- d. none of the above
- e. all of the above

The correct answer is b. A Nash Equilibrium is the best a player can do given another player's strategy.2. What type of strategy do rational players not choose to play?

- a. below dominant
- b. weakly dominant
- c. unplayable dominant
- d. strictly dominant
- e. absolutely dominant

The correct answer is d. A strictly dominant strategy is one that rational players do not play.3. In a two player game, the question of whether to advertise or not is asked. If both players advertise, they each earn a $5 payoff. If one advertises and the other does not, the one that advertises earns a $10 payoff and the other receives a $0 payoff. If neither advertises, they each earn a $3 payoff. Which of the following is (are) true?

- a. player 1 has a dominant strategy
- b. neither player has a dominant strategy
- c. player 2 has a dominant strategy
- d. there is a dominant strategy to not advertise
- e. both a and c

The correct answer is e. Both players have a dominant strategy to advertise.4. What term in game theory describes an action that remains constant regardless of an opponent's action?

- a. fixed
- b. dominant
- c. incorrect
- d. normal
- e. none of the above.

The correct answer is b. A dominant strategy is one where the best choice for a player is constant no matter the action another player takes.5. Two players earn $10 if they each do action A, $15 if they each do action B, $6 if one does action A and the other action B, and $3 if one does action B and the other action A. If player 1 chooses action B, what is the best possible choice for player 2?

- a. Action A
- b. Action B
- c. both a and b
- d. not enough information to tell
- e. no action

The correct answer is b. If player 1 chooses action B, player 2 can earn $6 or $15. Player 2 should choose to earn $15 by doing action B.ReferencesBaye, Michael R.,

"Managerial Economics and Business Strategy", McGraw-Hill Irwin. 2006.Dixit, Avinash and Barry Nalebuff.,

"Game Theory"."Dominance", a PowerPoint presentation."Dominance (Game Theory)", Wikipedia."Prisoner's Dilemma", Wikipedia.- http://en.wikipedia.org/wiki/Prisoner's_dilemma

Shor, Mikhael., Dictionary of Game Theory Terms, Game Theory.net"Solving Games: A Third Method"- http://isc.temple.edu/economics/Econ_92/Game_Lectures/4th-Nash/NashEq.htm

Stewart, Trudy.,"Students Learn Math 'Mind' Games", Stevens Point Journal, 1-9-02."Strictly Dominated Strategies", a PowerPoint presentation.External Linkshttp://mba651fall2007.wikispaces.com/Nash+equilibrium

http://mba651fall2007.wikispaces.com/prisoners+dilemma

http://mba651fall2007.wikispaces.com/sequential+move+games

Links for Funhttp://serendip.brynmawr.edu/playground/pd.html