🔎 Finding Dominant Strategies

Contents:

• Does Rob have a dominant strategy?
• Does Bruce have a dominant strategy?
• Dominant Strategy Equilibria in the Prisoner’s Dilemma
• Do all games have dominant strategies?

The prosecutor needs a confession to make the case. Therefore, they will give us a lighter sentence if we confess…

• If neither confess - ie if both of us stonewall the prosecution, then both only get 1 year in jail. (This is the ideal case if we could cooperate. Only 2 years spent in jail in total.)
• If both confess , they both go to jail for 5 years. (10 years spent in jail in total.)
• If one person confesses, the other one is sent in the slammer for 10 years, but the person who confessed goes free.

		Rob
		Don’t Confess	Confess
Bruce	Don’t Confess	$\pi_R = 1 \text{ yr}$ $\pi_B = 1 \text{ yr}$	$\pi_R = \text{Free}$ $\pi_B = 10 \text{ yrs}$
	Confess	$\pi_R = 10 \text{ yrs}$ $\pi_B = \text{Free}$	$\pi_R = 5 \text{ yrs}$ $\pi_B = 5 \text{ yrs}$

We Convert this to utilities. A higher utility means less prison time.

		Rob
		Don’t Confess	Confess
Bruce	Don’t Confess	$u_R = -1$ $u_B = -1$	$u_R = 0$ $u_B = -10$
	Confess	$u_R = -10$ $u_B = 0$	$u_R = -5$ $u_B = -5$

Beforehand, both suspects made a pact that they wouldn’t confess. Will they honor the pact?

(The connection to regular economics is that in an oligopoly, the two (or more) oligopolists would like to cooperate with each other and raise their mutual profits. However, this tends to be very difficult and they generally find it easier to compete with one another. As a result, oligopolies tend to be more efficient than monopolies. They can’t restrict output to raise prices the way that a monopoly can (or at least they can’t do it as effectively).

Does Rob have a dominant strategy?

For Rob, it is best to confess no matter what Bruce does. A dominant strategy is just a strategy that is best no matter what your opponent does. Therefore, confessing is Rob’s dominant strategy.

Is it better for Rob to Confess if Bruce Confesses?

		Rob
		Don’t Confess	Confess
Bruce	Don’t Confess	$u_R = -1$ $u_B = -1$	$u_R = 0$ $u_B = -10$
	Confess	$\boldsymbol{u_R = -10}$ $u_B = 0$	$\underline{\boldsymbol{u_R = -5}}$ $u_B = -5$

If Bruce confesses, I get -10 if I don’t confess and -5 if I do confess. Therefore, if Bruce confesses, I’m better off if I confess.

Is it better for Rob to Confess if Bruce doesn’t Confess?

		Rob
		Don’t Confess	Confess
Bruce	Don’t Confess	$\boldsymbol{u_R = -1}$ $u_B = -1$	$\underline{\boldsymbol{u_R = 0}}$ $u_B = -10$
	Confess	$u_R = -10$ $u_B = 0$	$\underline{u_R = -5}$ $u_B = -5$

If Bruce doesn’t Confess, I get -1 if I stay strong and don’t Confess and I get 0 if I turn my back on him and Confess.

A strategy is dominant if it is a best response no matter what the other player does. Therefore, Confessing is a dominant strategy for Rob.

If the numbers above represent my true utilities, then I will Confess. (We will generally assume that the numbers in the table represent your true utilities, and therefore include everything you Care about.)

Does Bruce have a dominant strategy?

For Bruce, it is better to confess, no matter what I do. Therefore, Bruce has a dominant strategy to confess. Let’s verify this:

Is it better for Bruce to Confess if Rob Confesses?

		Rob
		Don’t Confess	Confess
Bruce	Don’t Confess	$u_R = -1$ $u_B = -1$	$u_R = 0$ $\boldsymbol{u_B = -10}$
	Confess	$u_R = -10$ $u_B = 0$	$u_R = -5$ $\underline{\boldsymbol{u_B = -5}}$

If Bruce Confesses, he gets a utility of -5, but if he doesn’t Confess, he goes to jail for a very long time with a utility of -10. Therefore, if I confess, it is a ‘best response’ for Bruce to confess.

Is it better for Bruce to Confess if Rob doesn’t Confess?

		Rob
		Don’t Confess	Confess
Bruce	Don’t Confess	$u_R = -1$ $\boldsymbol{u_B = -1}$	$u_R = 0$ $u_B = -10$
	Confess	$u_R = -10$ $\underline{\boldsymbol{u_B = 0}}$	$u_R = -5$ $\underline{u_B = -5}$

If Bruce Confesses, he gets a utility of 0, but if he doesn’t Confess, he gets a utility of -1. therefore, if I don’t confess, it is a ‘best response’ for Bruce to confess.

Is Confession a dominant strategy for Bruce?

Confession is a dominant strategy for Bruce if it is better for Bruce to Confess no matter what Rob does.

Because it is better for him to Confess, no matter what I do, Confession is a dominant strategy for Bruce.

		Rob
		Don’t Confess	Confess
Bruce	Don’t Confess	$u_R = -1$ $\boldsymbol{u_B = -1}$	$u_R = 0$ $u_B = -10$
	Confess	$u_R = -10$ $\underline{\boldsymbol{u_B = 0}}$	$u_R = -5$ $\underline{u_B = -5}$

Dominant Strategy Equilibria in the Prisoner’s Dilemma

A dominant strategy equilibrium is when all players have a dominant strategy. Because both players have a dominant strategy, (Confess, Confess) is a dominant strategy equilibrium.

A dominant strategy is a “no-brainer” strategy because it is best no matter what the other player does. Therefore, if there is a dominant strategy equilibrium, we can be quite sure that it “will happen,” at least as long as the player’s utilities were specified correctly.

This is why we call it the prisoner’s dilemma. Both prisoners would far rather be in the equilibrium where neither of them Confess. In that equilibrium, we’d both get utilities of -1. However, because dominant strategies are dominant, both of us will actually Confess and we’ll both get utilities of -5.

This very powerful model illustrates how hard it can be to cooperate. It is useful in economics whenever two entities want to cooperate, but they end up pursuing their selfish ends, much to the detriment of both parties.

It was also very influential in the cold war to study arms races. Basically, both the soviet union and the US would prefer that we don’t have an arms race. However, both countries may consider it to be a dominant strategy to build up their arsenals. Thus, we may end up, unhappily building up our arsenals even though we wish we both weren’t doing so.

Do all games have dominant strategies?

No. Many don’t. Consider the following example. As is often done, the row player’s payoffs are listed first and the column player’s second. (Ron=Row; Cathy=Column.)

	Cathy → Ravel	Cathy → Chopin
Ron → Ravel	2,1	0,0
Ron → Chopin	0,0	1,2

(The underlines represent best responses for a given player.)

In this game, there are no dominant strategies.

For example, Ravel is not dominant for Ron because if Cathy goes to Chopin, Ravel is not optimal for Ron. Chopin is not dominant for Ron because if Cathy goes to Ravel, Chopin is not optimal for Ron.

Likewise, Cathy doesn’t have a dominant strategy.

The problem with dominant strategies is that many games will not have dominant strategies. For this reason, most game theorists use Nash Equilibrium as their solution concept most of the time.

✏️Does the following game have any dominant strategies?

	Cathy → Ravel	Cathy → Chopin
Ron → Ravel	2,1	0,0
Ron → Chopin	0,0	300,500

✔ Click here to view answer

Neither strategy has underlines all of the way across, so there are no dominant strategies for row.

Likewise, neither column has underlines all of the way down, meaning that neither column is optimal “no matter what the other player does.” Therefore, column doesn’t have a dominant strategy either.

	Cathy → Ravel	Cathy → Chopin
Ron → Ravel	2,1	0,0
Ron → Chopin	0,0	300,500

Both players might like to meet up at the Chopin concert, but if Cathy goes to Ravel, Ron wants to be there. Thus, Chopin isn’t a dominant strategy.  ✅