Nash Equilibrium
By Danny van der Steeg.


Introduction
Game theory, which is a very useful tool for managers, is the study of how decisions are made by people when the payoff they receive depends not only on their decisions, but also on the decisions of others. (McKenzie and Lee, 2006). The dominant strategy, prisoner’s dilemma and Nash equilibrium are a few examples of what is covered in Game theory. The topic of this paper is Nash Equilibrium.

I. History of Nash Equilibrium
The godfather of the concept Nash equilibrium is mathematician John Forbes Nash, who was born on June 13, 1928. He earned his doctorate in 1950 with his dissertation on non-cooperative games in which he formed the concept of the Nash equilibrium. Antoine Augustin Cournot elaborated on John Nash’s work and created the Nash equilibrium of the Cournot duopoly game. Before John Nash, John von Neumann and Oskar Morgenstern laid the groundwork for what we now call Nash equilibrium.

Fun facts about John Forbes Nash:
  • He attended college classes while still being in high school;
  • He got accepted into Harvard, but eventually went to Princeton;
  • The movie “A Beautiful Mind” with "Russell Crowe" was about him;
  • He received a Nobel Prize in Economics in 1994;
  • He suffered from schizophrenia.

II. Definition
A set of strategies constitute a Nash equilibrium if, given the strategies of the other players, no player can improve her payoff by unilaterally changing her own strategy (Baye, 2006). In other words, a set of strategies is a Nash equilibrium, when all players choose a strategy that uses their best response and no player can get a higher payoff by choosing a different strategy. An important aspect of Nash equilibrium is that each player has knowledge of the strategies of all other players. Every player is doing the best he or she can given this knowledge.

III. Theory
In order to explain Nash equilibrium, it is crucial to have knowledge on the concept dominant strategy. A dominant strategy is “a strategy that produces a higher payoff than any other strategy the player can use for every possible combination of its rival’s strategies” (Perloff, 2007). When a player in a game has a dominant strategy, it will choose that action regardless of what it’s rivals will choose. Dominant strategy is closely related to Nash equilibrium. Nash equilibrium is in fact a broader concept than dominant strategy. A Nash equilibrium does not have to have a dominant strategy, but if there is a dominant strategy, then there is a Nash equilibrium.

Game theory is a concept that is best explained with examples, therefore consider the following example:

Graph_1.JPG

Dominant strategy
In order to determine whether or not John and Achmed have a dominant strategy we have to analyze the game step by step.

Let’s start with John. Looking at the matrix given above from John’s perspective, he has two choices, which are black or white. Black has a yield of 8 or 16 and white has a yield of 4 or 0, both depending on whether Achmed chooses black or white.

  • If Achmed chooses black, John will choose black also, because that results in a yield of 8 instead of 4 when selecting white.
  • If Achmed chooses white, John will choose black, because that results into a yield of 16 instead of 0 when selecting white.

As can be concluded from the above analysis, John will always choose black despite of what Achmed will choose, because black will result in the highest payoff, which is either 8 or 16. Consequently, John’s dominant strategy is choosing black.

Now let’s proceed to see whether or not Achmed has a dominant strategy. Again, looking at the matrix given above from Achmed’s perspective, he has the same two options John has, black or white. Black has a yield of 24 or 28 and white has a yield of 20 or 40 depending on whether John chooses black or white.

  • If John chooses black, Achmed will choose black also, because that results in a yield of 24 instead of 20 when selecting white.
  • If John chooses white, Achmed will choose white also, because that results in a yield of 40 instead of 28 when selecting white.

Consequently, Achmed does not have a dominant strategy.

Nash equilibrium
Now that the concept of dominant strategy is clarified, it makes it easier to comprehend the concept of Nash equilibrium. As mentioned before, a situation constitutes a Nash equilibrium if given the strategies of the other players, no player can improve her payoff by unilaterally changing her own strategy.

As concluded in the section covering dominant strategy, John will always select black and therefore Achmed will select black because that will provide him with the highest payoff given the strategy of John. The strategies black/black is the Nash equilibrium, which results into the following yields: 8, 24.
Graph_1.JPG

Let’s make sure that neither of the players in the game cannot improve his payoff by unilaterally changing her strategy. The strategies black/black has already been identified as the Nash equilibrium.

  • If John decides to deviate to white from the black/black Nash equilibrium, his payoff will decrease to 4. John cannot improve his payoff.
  • If Achmed decides to deviate to white from the black/black Nash equilibrium, his payoff will decrease to 20. Achmed cannot improve his payoff.

Nash Equilibria
It is possible to have more than one Nash equilibrium in a game. See the following matrix for this:
Graph_2.JPG

If the same analysis is released on the above matrix, neither player A or B can unilaterally change her strategy to obtain a higher payoff at the cells in bold.

Let’s review the Nash equilibria from the matrix above to make sure neither of the players can obtain a higher payoff by unilaterally changing their strategy:
  1. (Blue, Yellow). Player A cannot obtain a higher payoff by switching to Yellow or Green. Player B cannot obtain a higher payoff by switching to Blue or Green.
  2. (Yellow, Blue). Player A cannot obtain a higher payoff by switching to Blue or Green. Player B cannot obtain a higher payoff by switching to Yellow or Green.
  3. (Green, Green). Player A cannot obtain a higher payoff by switching to Blue or Yellow. Player B cannot obtain a higher payoff by switching to Blue or Yellow.

Nash Bargaining
Bargaining is another aspect that comes up when discussing Nash equilibria. When bargaining in a simultaneous, one-shot bargaining game, the players have only one chance to reach an agreement and the offers are made simultaneously. Suppose management and a labor union are bargainig on how much of $100 profits should be given to the union. The $100 dollars can only be split up in parts of $50 dollars. The players have one chance to reach an agreement and they submit the amount they would like simultaneously (a simultaneous, one-shot bargaining game). Each player can choose either 0, 50, or 100 dollars. If, the sum of the amounts of the two parties together exceeds $100 then there is no deal made and this will cost both the union and management $1. See the following matrix (Baye, 2007, p.364).

nash_bargaining.JPG

If we analyize the matrix of payoffs, it can be determined that there are three Nash equilibria.
1) The first Nash equilibrium is (100,0) where management asks for 100 and the union for 0.
2) The second Nash equilibrium is (0,100) where management asks for 0 and the union for 100.
3) The third Nash equilibrium is (50,50) where both management and union ask for 50.

At the three combinations of payoffs, neither the union or management can unilaterlly improve their payoffs.

Both the first and third combination are unlikely to happen, because in those combinations either the management or the union will have to choose 0, which is unlikely. Therefore, it is not wise for the management or the union to choose 100, because it is likely that the other party will choose 50 or 100. This will result in a cost of 1 dollar for each party, because the total amount asked will exceed 100. The second combination is (50,50) where each party receives an equal share.

It is difficult to predict the outcome of a simultaneous-move bargaining game, because there are multiple Nash equilibria. This can result into inefficiencies when the parties do not coordinate on an equilibrium. In the above matrix, six of the nine outcomes are inefficient, because it results in a total payoff that is less than 100. Three of these result in losses (real world example: failing of an agreement). According to Baye, "experimental evidence suggests that bargainers often perceive a 50-50 spilt to be fair. Consequently, many players in real-world settings tend to choose strategies that result in such a split even though there are other Nash Equilibria." In the above example, the most likely Nash equilibrium is for the union and the management to each ask for $50.

The concept of collusion is only briefly covered. Collusion depends on a number of factors. To read more about collusion, finitely and infinitely repeated games, click here .

Prisoner's dilemma
Prisoner's dilemma is another concept related to game theory. This concept will only be discussed briefly, because the main topic of this paper is Nash equilibrium. A more indepth explanation of prisoner's dilemma can be found here. The game shown earlier with John and Achmed is an example of a game that is not a prisoner's dilemma game. It is not one, because there is no other choice that would make both players better off. If there was a choice that would make both player better off, this would be a prisoner's dilemma

IV. Real world application
Let’s take a hypothetical situation in which Apple and Microsoft are the only two producers of desktop computers. They both have an identical product that they would like to introduce into the market. One decision that they have to make concerns the price of their product; will they choose a high price or a low price. In order to simplify this situation, let us assume that the high price is 1500 dollars, the low price is 900 dollars and the amounts given in the matrix are profits in dollars.
AppleMicrosoft.JPG

The first step in determining which pricing strategies both companies will use in the market is seeing if Apple or Microsoft has a dominant strategy.

Does Microsoft have a dominant strategy?

• If Apple chooses a high price, Microsoft will choose a low price, because they will prefer to make a profit of 1200 dollars over a profit of 600 dollars.
• If Apple chooses a low price, Microsoft will choose a low price, because they will prefer to make a profit of 900 dollars over a profit of 300 dollars.

Microsoft will choose a low price no matter what price Apple chooses and therefore Microsoft has a dominant strategy, which is a low price.

Does Apple have a dominant strategy?

• If Microsoft chooses a high price, Apple will choose a low price, because they will prefer to make a profit of 1200 dollars over a profit of 900 dollars.
• If Microsoft chooses a low price, Apple will choose a high price, because they will prefer to make a profit of 1200 dollars over a profit of 900 dollars.

Apple chooses a low price if Microsoft chooses a high price and a high price if Microsoft chooses a low price, and therefore Apple does not have a dominant strategy.

Knowing whether or not Apple and Microsoft have a dominant strategy makes it easy to determine which pricing strategies both companies will use. Microsoft’s dominant strategy is to choose a low price and the previous analysis has shown that Apple will choose a high price. This results into a profit of 1200 dollars for Microsoft and Apple each. This is the Nash equilibrium.

By checking whether or not Apple or Microsoft can change their strategy to improve their payoffs, it is possible to make sure that the pricing strategies (low, high) is a Nash equilibrium. Remember, a situation constitutes a Nash equilibrium if given the strategies of the other players, no player can improve her payoff by unilaterally changing her own strategy. If the above matrix is reviewed, it can be concluded that neither Apple or Microsoft can unilaterally change their strategy to improve their payoff.

V. Questions

Question 1.
Which condition must be true in order for a set of strategies to constitute a Nash equilibrium?

A) If, given the strategies of the other players, at least one player can improve her payoff by unilaterally changing her own strategy
B) If, given the strategies of the other players, every player can improve her payoff by unilaterally changing her own strategy.
C) If, given the strategies of the other players, no player can improve her payoff by unilaterally changing her own strategy.
D) There is no such thing as Nash equilibrium.

Use the following matrix for question 2, 3, 4 and 5.

Graph_3.JPG
Question 2.
What is the Nash equilibrium of this game?

A) Left, Right
B) Right, Right
C) Left, Left
D) Right, Left

Question 3.
Is there a different choice either of the firms can choose that would make them both better off? If so, what would that be?

A) Yes, player A chooses Right and player B chooses Left.
B) Yes, player A chooses Left and player B chooses Right.
C) No, there is no choice either of the firms can choose that would make them better off.
D) Yes, both players choose Right.

Question 4.
Do player A and B have a dominant strategy? If so, what would it be?

A) Player A has a dominant strategy: Right
B) Player B has a dominant strategy: Right
C) Neither player has a dominant strategy.
D) Both players have a dominant strategy: Left

Question 5.
At Nash equilibrium, how much profit does player A make and how much profit does player B make?

A) Player A: $20 million, Player B: $20 million
B) Player A: $17 million, Player B: $25 million
C) Player A: $22.5 million, Player B: $22.5 million
D) Player A: $20 million, Player B: $22,5 million

Answers
1. C: If given the strategies of the other players, no player can improve her payoff by unilaterally changing her own strategy. This is the condition that is true. See the definition of Nash equilibrium.
2. C: Left, Left is the Nash equilibrium. When both players choose Left, no player can unilaterally change her strategy to obtain a higher payoff.
3. D:Yes, both players choose Right. If both players will choose right they each make 22.5. Note: the players cannot get to that point by unilaterally changing their strategy. Both players will have to choose right.
4. D: Both players have a dominant strategy: Left. Both players will select Left no matter what the other player’s strategy is in order to maximize their payoff.
5. A: Player A: 20, Player B: 20. Nash equilibrium is when both players select Left and with those strategies, the payoff for both players is $20 million.

VI. References