Lecture Notes 7: Examples and Discussion of Simultaneous Games
Thus far, we have developed quite a bit of theory on solving simultaneous games. In this section we will go through some important examples of simultaneous games, discuss and extend our equilibrium concepts, and finish with a brief discussion of some evidence on how well game theory predictions pan out in experiments and in the real world.
Dominant Strategies
Economists consider dominant strategies to be among the most convincing of all game theory predictions. Consider the following example, showing sales revenues of two newspapers depending upon which story they highlight on the front page.
AIDS Budget
AIDS 35,65 70,30
Budget 30,70 45,55
Running the AIDS story is a dominant strategy for the first player. No matter which story the second player runs, the first player always gets a higher payoff by running the AIDS story. Running the AIDS story is also a dominant strategy for the second player. The dominant strategy equilibrium, where both newspapers run the AIDS story, is a very convincing prediction of equilibrium play. If both companies are better off running the AIDS story no matter what the other company does, then it’s obvious that this will be the outcome.
Even if only one player has a dominant strategy, the equilibrium prediction is still pretty convincing. Consider the following modification of the payoffs given above.
AIDS Budget
AIDS 42,58 70,30
Budget 30,70 18,82
Player 1 has a dominant strategy to run the AIDS story. Player 2 does not have a dominant strategy (running the budget story would be better if player 1 runs the budget story), but we assume common knowledge of the setup of the game. It’s not too much of a leap to say that player 2 should realize that player 1 will always run the AIDS story, and then she will run the AIDS story as well.
fixed at 100, any improvement in one player’s payoff corresponds exactly to a decline in the other player’s payoff.
People who don’t understand game theory frequently mischaracterize the concept of a dominant strategy. It doesn’t mean that you’re always better off than your opponent, and it certainly doesn’t mean that it’s the highest payoff in the game. A dominant strategy just means that for each possible strategy chosen by your opponents, your best response is always the same strategy.
Example of Dominant Strategies
A corporate raider is trying to buy out a majority of shares of stock in a company. Each share is currently worth $100. There are 100 shareholders, each with one share. The raider takes over the company if he can capture 51 or more of the shares. If he does take over, he has a right to seize any of the remaining shares for $90. The raider makes an interesting deal, after which all the shareholders have to decide simultaneously what to do.
• If between 0 and 50 people agree to sell, the raider will pay $105 per share.
• If 51 or more people agree to sell, the raider will pay $95 per share.
The raider’s offer makes it a dominant strategy for each shareholder to sell. If 50 or fewer people agree to sell, then each shareholder is better off with the $105 from selling than with the $100 from not selling. But if 51 or more of your fellow shareholders do agree to sell, then any individual shareholder is better off with the $95 that the raider offers versus the $90 that he can seize it at after taking over. If you are an individual stockholder, regardless of what your fellow stockholders do, you are always better off selling. Everyone plays a dominant strategy and sells.
The irony here is that everyone selling lowers the value of their shares from $100 to $95. If the shareholders could coordinate and all agree not to sell to the raider, they would all be better off. But, given the raider’s “dirty trick”, each shareholder is better off individually by selling, regardless of what the other shareholders do.
Some Important Games
There are a few games that come up a lot. It is worth being familiar with them because they are often used as canonical examples of different incentive patterns.
Coordination Game
Java C++
Java 1,1 0,0
C++ 0,0 1,1
The important thing is that both play the same strategy. (Java, Java) and (C++, C++) are both Nash Equilibria, and the concept of Nash Equilibrium doesn’t give us any help in settling on one equilibrium or the other. Both would be mutual best responses. Even worse, there is also a mixed equilibrium where both players randomize over the two options with equal probability. The average payoffs in this equilibrium are lower than the payoffs for either pure strategy equilibrium, and in fact there is a 50% chance of not matching strategies at all in the mixed equilibrium!
Assurance Game
Two engineers are working on a project and can design it using Metric or British units. This is also a coordination game, although it is better in this case for them to coordinate on Metric.
Metric British
Metric 2,2 0,0
British 0,0 1,1
(Metric, Metric) and (British, British) are both Nash Equilibria. Unfortunately, the (Metric, Metric) outcome is not assured. The strategy set where both use British units is still a Nash Equilibrium, and in fact Metric is not even a dominant strategy – strictly or weakly. If the other player chooses British, it is your strict best response to also use British units. The concept of Nash Equilibrium doesn’t help us get to the better (Metric, Metric) outcome.
Battle of the Sexes
Two partners (with no mobiles!) are driving to a date, but never set a firm place. They had discussed boxing and the opera. The boyfriend (player 1) prefers boxing and the girlfriend (player 2) prefers the opera, but not meeting up is the worst of all.
Boxing Opera
Boxing 2,1 0,0
Opera 0,0 1,2
Chicken
Two drivers are playing chicken and speeding towards each other. Each one can swerve or keep going straight. You lose face if you swerve while the other person continues going straight, but the worst outcome of all is if both go straight and they crash.
Swerve Straight
Swerve 0,0 -1,1
Straight 1,-1 -2,-2
There are two pure strategy equilibria: (Swerve, Straight) and (Straight, Swerve). But again the concept of Nash Equilibrium gives us no reason to prefer one over the other. How could you get your preferred outcome here? Giving the impression that you are committed to going straight is the key to making the other player swerve. In a later unit, we will talk extensively about the idea of credibility in commitments. There is also a mixed strategy equilibrium.
Prisoners’ Dilemma
Police are interviewing two criminals, whom they suspect of committing a crime, in separate rooms. If both confess, both will get sentenced to 10 years in jail. If both deny the crime, the police eventually have to release them, and both will spend only 2 years in jail.
The police offer a deal to each suspect: If you confess, but your partner denies the crime, you can go free right now but your partner will have to spend 30 years in jail. Of course, if your partner confesses and you keep denying it, then he will go free and you have to spend 30 years in jail. The setup of this game is as follows.
Deny Confess
Deny 2 years jail, 2 years jail 30 years jail, Free
Confess Free, 30 years jail 10 years jail, 10 years jail
Confessing is a strictly dominant strategy for both players. If the other suspect denies the crime, you can go free instead of spending 2 years in jail by confessing. If the other suspect confesses, you can spend 10 years in jail instead of spending 30 years in jail by confessing. No matter what the suspect in the other interview room does, you are better off by confessing. The same can be said for the other suspect. The only Nash Equilibrium is for both players to confess.
The prisoner’s dilemma is very important and there are a whole plethora of economic problems that have the structure of a prisoner’s dilemma.
• Oligopoly pricing problems – Both firms are better off if both maintain a high price, but this is not an equilibrium. If your rival maintains a high price, you can do even better by undercutting him and stealing his customers.
• Public goods – Society is better off if everyone contributes a little bit to provide public goods, but each individual is better off being a “free-rider” and not making any contributions.
It seems like there should be some way to solve this problem and “resolve” the prisoner’s dilemma. There is an extensive body of literature on this issue in game theory, and we will go through some of the ideas later in the semester. Most notably, the problem could be solved in a repeated game where each player has an incentive not to defect in order to build a long-run reputation and encourage cooperation from other players.
Focal Points
In a game with multiple equilibria, what might make one “stick out” from the others? A focal point is some solution that people will tend to in the absence of explicit coordination.
As a simple example, suppose that you and a friend agree to meet at 2 p.m. tomorrow but you never agreed on the place and you can’t communicate with each other. You could probably both imagine a favorite restaurant or some place that both of you would realize should “stick out” from all the possibilities. This is effectively a coordination game.
As another example, think about this game.
A B C D
A 10,10 0,0 0,0 0,0
B 0,0 10,10 0,0 0,0
C 0,0 0,0 9,9 0,0
D 0,0 0,0 0,0 10,10
For a final example, suppose that two players are given a slip of paper on which they write down a number. If the numbers add up to exactly 100, both players get an amount of money equal to the number they wrote down. If the numbers do not add up to 100, both players get nothing.
This game has many Nash Equilibria, e.g. (50,50) and (70,30). Even something like (1,99) is a Nash Equilibrium. If you are the $1 player, you might be unhappy, but given that the other player writes down $99 you are still better off with $1 than by writing down anything other than 1 and ending up with nothing. However, out of all these Nash Equilibria, if there is no communication then the obvious focal point of this game is probably (50,50).
Two Theoretical Results on Nash Equilibrium
Here are two important, general results on Nash Equilibrium.
• Existence of Nash Equilibrium: Every game with a finite number of players and a finite number of strategies has at least one Nash Equilibrium.
• If finite, the number of Nash Equilibria (including pure and mixed equilibria) in any
non-degenerate game is odd.
The first is a seminal result in game theory proved by John Nash in 1950. Any game with a finite number of players and a finite number of strategies has at least one Nash Equilibrium. Of course, it might be a mixed equilibrium rather than a pure equilibrium. The theorem breaks down for games with infinite numbers of players or games with strategy sets that are not finite. Those games might not have a Nash Equilibrium.
The second result is a special case of a general result in topology. In non-degenerate games (no ties), the number of Nash Equilibria will always be odd, not even.
Refinements of Nash Equilibrium
In games with multiple equilibria, game theorists are sometimes interested in narrowing down the set of Nash Equilibria by developing equilibrium refinements, which are further restrictions beyond the definition of Nash Equilibrium that might eliminate some of the Nash Equilibria. Two of the most popular are as follows.
Consider the following game.
L R
T 5,5 4,2
B 2,4 4,4
(𝑇𝑇, 𝐿𝐿) and (𝐵𝐵, 𝑅𝑅) are both Nash Equilibria. Let’s examine (𝑇𝑇, 𝐿𝐿). 𝑇𝑇 is a strict best response for player 1 when player 2 plays 𝐿𝐿. Similarly, 𝐿𝐿 is a strict best response for player 2 when player 1 plays 𝑇𝑇. This means that (𝑇𝑇, 𝐿𝐿) is a strict equilibrium.
On the other hand, consider the equilibrium (𝐵𝐵, 𝑅𝑅). When player 2 plays 𝑅𝑅, then 𝐵𝐵 is only a weak best response for player 1, since 𝑇𝑇 and 𝐵𝐵 both provide the same payoff for player 1. Similarly, 𝑅𝑅 is only a weak best response for player 2 when player 1 plays 𝐵𝐵.
Summarizing, although (𝑇𝑇, 𝐿𝐿) and (𝐵𝐵, 𝑅𝑅) are both Nash Equilibria, only (𝑇𝑇, 𝐿𝐿) is a strict equilibrium because 𝑇𝑇 and 𝐿𝐿 are strict best responses for the players using them.
Don’t make the mistake of thinking that (𝑇𝑇, 𝐿𝐿) is a strict equilibrium because the payoffs are higher than the payoffs for (𝐵𝐵, 𝑅𝑅). Here is another example.
Y Z
A 3,3 1,1
B 3,1 2,2
In the this game, (𝐴𝐴, 𝑌𝑌) and (𝐵𝐵, 𝑍𝑍) are both Nash Equilibria. But (𝐴𝐴, 𝑌𝑌) is not a strict equilibrium because 𝐴𝐴 is not a strict best response for player 1 to player 2 playing 𝑌𝑌 – 𝐴𝐴 and 𝐵𝐵 are equally good choices for player 1 in this case. On the other hand, (𝐵𝐵, 𝑍𝑍) is a strict equilibrium since 𝐵𝐵 is player 1’s strict best response to 𝑍𝑍 and since 𝑍𝑍 is a strict best response by player 2 to 𝐵𝐵.
• A perfect equilibrium in a 2-player game is a Nash Equilibrium that does not involve any weakly dominated strategies by either player.
Consider the following game.
G B
G 1,1 0,0
In this game, (𝐺𝐺, 𝐺𝐺) and (𝐵𝐵, 𝐵𝐵) are both Nash Equilibria. However, 𝐵𝐵 is a weakly dominated strategy for each player – Either worse than 𝐺𝐺 or tied with 𝐺𝐺 depending upon the opponent’s choice. This means that (𝐵𝐵, 𝐵𝐵) is not a perfect equilibrium. Even though it is a Nash Equilibrium, it involves weakly dominated strategies.
The idea behind perfection is to imagine a chance that the other player will make a mistake. If there’s even a slight chance that the other player will play 𝐺𝐺, then you are better to play 𝐺𝐺 also. It’s never worse than 𝐵𝐵, but sometimes it’s better.
Again, don’t make the mistake of thinking that the perfect equilibrium will always be the equilibrium with higher payoffs. Consider the following game.
Y Z
A 1,1 2,0
B 0,2 2,2
In this game, (𝐴𝐴, 𝑌𝑌) and (𝐵𝐵, 𝑍𝑍) are both Nash Equilibria, but only (𝐴𝐴, 𝑌𝑌) is a perfect equilibrium. This is because 𝐵𝐵 is weakly dominated by 𝐴𝐴 for player 1 and 𝑍𝑍 is weakly dominated by 𝑌𝑌 for player 2.
This characterization of perfection works only for 2-player games. There is a more technical definition of perfect equilibrium for games with a larger number of players.
Equilibrium Refinements – Example
Let’s find all the Nash Equilibria, strict equilibria and perfect equilibria of the following game. Best responses are bolded. We will restrict our attention to pure strategies.
W X Y Z
A 6,3 6,6 1,1 1,7 B 6,3 5,4 4,5 1,0
C 6,3 3,2 3,2 0,2
Using standard best response analysis, there are three pure-strategy Nash Equilibria: (𝐴𝐴, 𝑍𝑍), (𝐵𝐵, 𝑌𝑌) and (𝐶𝐶, 𝑊𝑊).
To find the perfect equilibria, we need to figure out if there are any weakly dominated strategies. Player 2 does not have any weakly dominated strategies. But notice that 𝐶𝐶 is weakly dominated by 𝐵𝐵 for player 1. It’s either a tie or worse than 𝐵𝐵, for any choice that player 2 might make. Thus, the game’s perfect equilibria are (𝐴𝐴, 𝑍𝑍) and (𝐵𝐵, 𝑌𝑌). The other Nash Equilibrium, (𝐶𝐶, 𝑊𝑊), is not a perfect equilibrium because it involves a weakly dominated strategy for player 1.
Rationalizing Nash Equilibrium
A Nash Equilibrium is a set of mutual best responses. But if a game is simultaneous, why does it make sense to think of best responses to things that haven’t yet happened?
Well, we all choose strategies blindly sometimes. We put ourselves “in the other person’s shoes” and assume that the other person is going to do the same thing in trying to guess our behavior. Precisely, then, Nash Equilibrium really incorporates a set of mutual beliefs. The concept of a best response makes sense in this context – we aren’t responding to as-of-yet unobserved actions but to beliefs. Of course, uncertainty about these beliefs is allowed, which gives us mixed strategies.
Basically, Nash Equilibrium is a kind of stability condition. If we find ourselves at a Nash Equilibrium, then there is no incentive for unilateral deviation by any player.
Laboratory Experiments and Evidence
Laboratory experiments are increasingly popular in game theory – bring in subjects, provide monetary incentives and see what they do. To what extent does do laboratory experiments “test” the accuracy of equilibrium predictions provided by game theory?
The concept of using laboratory experiments to “test” game theory remains controversial among game theorists. Opponents argue that game theory is basically a set of tautologies (assertions that hold by definition) and that you can never really “test” game theory. There are always a whole host of outside factors that make laboratory experiments an imperfect representation of a game that’s written down on paper. Nevertheless, experimental economics is becoming an increasingly accepted tool in game theory, and in fact experimental economists (Daniel Kahneman and Vernon Smith) won the Nobel Prize in economics in 2002.
Focal Points
Kreps (1980) did an interesting experiment where he assigned Student A “Boston” and assigned Student B “San Francisco”. Each was then separately given a list of some other cities and told to divide the cities up between the two students. If the students divided the cities in the same way, both won a prize. This is a coordination game with many Nash Equilibria. The assignments just have to match.
What do you think happened? Almost 80% of US residents won the prize. The obvious focal point for people who understand US geography is that the students assigned cities on or near the East Coast to Student A and cities on or near the West Coast to student B. For non-US residents there is no obvious focal point, and results were hit-and-miss. Some of them tried to split the cities up alphabetically, for example.
Altruism
There is a consistent willingness to cooperate in public goods and prisoners’ dilemma type games, despite the unequivocal strategic problem. This is a hot topic in the literature, and we will come back to this a few times.
Experience
Here’s a simple game. Each player writes down a number from 0 to 100. The winning number is half of the average. The only Nash Equilibrium is for everyone to write down 0.
What do you think happened? Many people aren’t good at math and wrote down 25 or 50 in the first round, but within two or three rounds people figured out what was going on and the outcome quickly converged to everyone writing down something pretty low. Interestingly, watching others play is even better than playing yourself.
Baseball
An interesting “real-life” application outside of the laboratory is baseball. Baseball has been around commercially in the US for more than 100 years.
Mixed Strategies: Changing Mixture Probabilities
There is an interesting property of mixed equilibria. Consider the following zero-sum game.
X Y
A 30,70 60,40
B 90,10 20,80
The Nash Equilibrium is (0.7𝐴𝐴 + 0.3𝐵𝐵, 0.4𝑋𝑋 + 0.6𝑌𝑌).
Now let’s change this slightly to make player 1 more successful when he plays 𝐴𝐴 against player 2 playing 𝑌𝑌. Everything else is the same.
X Y
A 30,70 65,35
B 90,10 20,80
The Nash Equlibrium of this new game is (0.67𝐴𝐴 + 0.33𝐵𝐵, 0.43𝑋𝑋 + 0.57𝑌𝑌).
Even though the only change in the game is to make 𝐴𝐴 a more attractive option for player 1, he actually plays it less often in the new equilibrium! What’s going on here? Note that player 2 is going to respond to this change by playing 𝑋𝑋 more often, in which case player 1 actually prefers to put more weight on 𝐵𝐵.
This phenomenon is common in sports. If you get really good at hitting a certain shot in tennis, you might be less likely to use this shot in equilibrium. If your opponent knows that you are really skilled with a certain shot, he’ll adjust so as not to set you up for it very often and will become more effective at blocking you. It might pay in the end to use it less often.
Mixed Strategies vs. Correlated Strategies
There is a subtle point about mixed strategies that people sometimes confuse. Consider the following game.
X Y
A -50,-50 0,10
B 10,0 1,1
But this requires what we call correlated strategies – the players need access to some device that correlates their randomization. Normal mixed strategies where the players mix independently will not work to accomplish this. After all, if player 1 mixes 𝐴𝐴 and 𝐵𝐵 independently of player 2 mixing 𝑋𝑋 and 𝑌𝑌, then sometimes they will end up at (𝐴𝐴, 𝑋𝑋)! They need a way to coordinate their randomization. We will not go into any more detail about this topic, but it is important to note that independent mixtures are not the same as correlated randomization.
Discussion and Evidence on Mixed Strategies
Some game theorists think that mixed strategies are unattractive as a practical prediction of real-world strategy choices for a few reasons.
• Mixed strategy equilibria are not “stable” – they are extremely sensitive to mixture probabilities. The probabilities have to be exactly right. With a tiny perturbation of the probabilities, players would strictly prefer one pure strategy or the other.
• Mixed strategy equilibria are always weak Nash Equilibria meaning that players are indifferent over the strategies used in the mixture. This is exactly how mixed equilibria are constructed. Nash Equilibria are usually more convincing as a real-life prediction of play if the strategies constitute a strict best response.
• A player’s strategy choices in a mixed-strategy equilibrium are actually designed to make his opponent indifferent. For example, player 2’s mixture probabilities are determined by the condition that player 1 be indifferent between his strategies. Player 2’s mixture probabilities are independent of player 2’s own payoffs. Does it really make sense that people choose strategies like this?
According to Charles Holt, an experimental economist:
Subjects in experiments are rarely (if ever) observed randomizing, and when told afterwards that the equilibrium involves randomization, subjects express surprise and skepticism.
Nevertheless, there are a few well-known examples for which mixed strategies provide a convincing interpretation of behavior.
joint attack) or whether to drive dispersed (better against snipers). Playing one strategy or the other all the time was no good because the enemy could respond in kind. Instead, every morning, the General flipped a coin to decide how to send the trucks. This is a true application of a mixed strategy, with precisely the motivation to which we ascribe mixed strategies in game theory. Keep your opponent guessing.
This is a good example. What does randomization mean, anyway? Predictably alternating each day between the two strategies won’t work and is not randomization. Your opponent would always know how to respond. It doesn’t matter if your opponent knows that you are randomizing or the probabilities with which you are randomizing (indeed, the probabilities are chosen to make your opponent indifferent). But your opponent cannot find out how you are randomizing. You need to keep him guessing. The General found a good way to do that. Each day, the strategy he used was a surprise even to him.
• Mixed strategies are also common and clear in many sports examples. Choosing a strategy for each play in tennis or a penalty shot in soccer is effectively a simultaneous game. In fact, economists have run the numbers and there is very good evidence that players mix almost exactly according to what theory would predict.
An important point about these mixtures is that tennis players sometimes serve in a way that isn’t their best shot, and soccer players sometimes kick with the “wrong” foot just to protect their reputation of mixing. If you only watch one play, it might seem stupid to you that a tennis player doesn’t use her best shot or that a soccer player doesn’t kick with his best foot, but in the context of mixing it makes perfect sense. There will be times after the fact when a choice seems bad, but the more relevant question is whether it makes sense overall in the context of all the games that the player plays.
• During World War II, the British realized that there was a German spy at their staff meetings. Rather than arrest him, they started giving out false information at the meetings. The Germans eventually caught on. On D-Day, the British actually announced the correct location of the landing, knowing that the Germans would assume that this was false information. If mixed strategies are being used by all sides, then you might as well just ignore what the other person says!
What about business applications? It’s often difficult to convince corporate types to leave outcomes to chance. High-paid managers usually think they’re better at making decisions than a coin is. Nevertheless, there are a couple of examples that seem relevant.
last minute is predictable, then a lot of people might just plan on waiting until the last minute to buy them.
Problems
1. Consider the game given below. Restrict your attention to pure strategies.
a. Find the Nash Equilibria. b. Find the strict equilibria. c. Find the perfect equilibria.
L C R
T 3,5 2,3 1,1
M 3,1 3,2 4,5
B 3,4 1,3 4,4
2. Nash (1950) proved that all games with a finite number of players and a finite number of strategies have at least one Nash Equilibrium. But this result may not hold for games with an infinite number of players or with an infinite strategy set. Explain why each of these games has no Nash Equilibrium.
a. (Infinite set of players) An infinite set of players decide simultaneously whether to choose one of two strategies: 𝐼𝐼𝐼𝐼 or 𝑂𝑂𝑂𝑂𝑂𝑂. Any player choosing 𝑂𝑂𝑂𝑂𝑂𝑂 gets a payoff of 0. Any player who chooses 𝐼𝐼𝐼𝐼 gets a payoff of 1 if a finite number of players choose 𝐼𝐼𝐼𝐼. But any player who chooses 𝐼𝐼𝐼𝐼 gets a payoff of −1 if an infinite number of players choose 𝐼𝐼𝐼𝐼.
b. (Infinite strategy set) Firm 1 can produce output at a cost of $1.00 per unit and firm 2 can produce output at a cost of $0.90 per unit. Each firm chooses what price to charge. If the firms charge different prices, all the customers go to the firm with the lower price. If the firms charge the same price, they split the customers evenly.
3. It is generally true that the number of Nash Equilibria is odd (counting both pure and mixed equilibria). But this result can sometimes fail to hold. Show that the game below has two pure-strategy equilibria and no mixed-strategy equilibrium, violating the general principle.
L R
T 0,0 0,0
