Game Theory and Applications: an Analysis

Introduction:

I first learned about Game Theory during my Economics class, as an introduction to oligopolies and cartels. Ever since, I’ve been fascinated by the prisoner’s dilemma – how sometimes the most logical decision isn’t the option with the highest pay-offs. After researching about it, I also discovered other forms of Game Theory such as Hawk-Dove and Zero-Sum game. When our Economics teacher ran the simulation, I was surprised at how much our class deviated from what was deemed rational. After that lesson and watching “A Beautiful Mind”, I wanted to investigate the mathematics behind it.

Game theory is the use of mathematical modeling and analysis to describe and predict economic and psychological behavior. It is mostly used to make decisions in scenarios of conflict involving at least two players, where the yield of a player will depend on other players. Therefore, his actions are fully based upon his judgment about what other players will decide on – there is interdependence. Aside from economics, it can also be applied in biology, computer science, political science and psychology.

According to MIT, Nash Equilibrium occurs when “players guess the other players’ strategies and choose the most rational option available”. Rationality, therefore, is when a player maximizes his utility and payoffs. When Nash Equilibrium is achieved, there is no incentive to change – it is stable. In the exploration, the Nash Equilibrium is highlighted for each scenario.

This exploration will consider 3 types of game theory (Prisoner’s Dilemma, Hawk-Dove and Zero-sum) by looking at 6 scenarios: the basic prisoner’s dilemma, cartels, entering a monopolistic market, investing in technology depending on firm size and determining where to locate. To investigate the reliability of Game Theory in predicting real-life behavior, I also carried out a survey that covered 4 of the scenarios above – the basic prisoner’s dilemma, investing in technology depending on two firm sizes and locating.

Prisoner’s Dilemma

Prisoner’s dilemma is the most popular example of game theory. This game includes at least two players and imperfect information. It can be applied in many fields such as economics, psychology, biology and politics. The scenario I used in the survey is given below:

Example 1: You (A) and a friend (B) are arrested for a crime and sentenced to 2 years. Both of you are also suspected for a much larger crime (which you didn’t do), but the police do not have enough evidence to prove it. You are in solitary confinement with no means of speaking with each other. They give you a bargain – you could confess to committing the larger crime, betraying your friend, or deny it. These sentences are modeled in the pa-off table below – Figure 1 shows the number of years you will receive for each possible outcome (ie if you confess and they deny you get 1 year and they get 10 years)

At first, both choosing to deny seems like the most optimal yield (2,2). However, this is not the case; for both players, there is an incentive to confess. If player A confesses, he will either receive 5 years if B also confesses or 1 year if B denies. However, if A denies, he will either receive 10 years if B confesses or 2 years if B denies. The most rational decision would be to select the option with the least repercussion – to confess. This is because the worst possible scenario if A confesses will be 5 years, while the worst if A denies would be 10 years. This also applies to player B. Another incentive to confess is the fear of being betrayed, thus receiving 10 years, which is the worst possible scenario. This reasoning is called backward induction. Though (2,2) appears to be the most optimal yield, in this game, if the players were rational, they would both choose to confess (5,5). This stable payoff is called Nash Equilibrium.

It can be deduced that there are 4 possible combinations from the table above, and using combinations.

A= (_1

2)C

A=2

B= (_1

2)C

B=2

A×B=4

A general formula for prisoner’s dilemma can be found in the payoff table below, where w > x > y > z (w is most favourable (when one betrays the other), x is the most optimal yield (when both cooperate), y is the Nash equilibrium and z is least favourable (when one is betrayed by the other). In all cases, it would be best to confess.

Figure 2:

A

Confess Deny

B Confess y, y z, w

Deny w, z x, x

Prisoner’s dilemma has many applications in Economics. For example, it can be used to make decisions on whether a firm should invest in technology, advertising, R&D, join a cartel etc given the size of its competitor.

Example 2: You are Firm A in a cartel with Firm B. You are the same size as Firm B. You have to decide whether to follow the set price of the cartel or lower your prices (thus benefitting you), betraying the rules of the cartel. The payoff tables below show the consequences of all possible outcomes on existing profits.

Figure 3:

A

Follow Cheat

B Follow $20m, $20m $50m, -$10m

Cheat -$10m, $50m $0, $0

Figure 3 displays both firms in the short run. Following the backward induction reasoning used above, even though following the cartel agreement seems the most Nash Equilibrium is achieved when both firms cheat – the worst possible payoff is -$10million if Firm A follows and $0 if Firm A cheats (also applicable to Firm B). Furthermore, there is temptation to cheat; the best possible payoff is $20m if Firm A follows and $50m if Firm A cheats.

A general formula for the payoff table above can also be found in the table below, where w>x>y>z.

Figure 4:

A

Follow Cheat

B Follow x, x w, z

Cheat z, w y, y

However, if this were the case, there wouldn’t be cartels in the world. In reality, there would be retribution for cheating a cartel in the form of a price war. Figure 5 below displays both firms in the long run after a price war due to retribution (which reality and economic theory claim will occur when at least one firm cheats).

Figure 5:

A

Follow Cheat

B Follow 20, 20 -∞, -∞

Cheat -∞, -∞ 0, 0

Because Firm A is equal in size to Firm B, a price war would be mutually destructive, leading to a payoff of -∞. As the worst possible payoffs are equal, the highest possible payoff will determine whether a firm should remain in the cartel or cheat. As $20 million is higher than $0, it is in both firms’ interest to remain in the cartel, contrasting the earlier conclusion derived from the short-run payoff table. A general formula can also be found in the table below, where x>y>t.

Figure 6:

A

Follow Cheat

B Follow x, x t, t

Cheat t, t y, y

In conclusion, when there is retribution, it would be best to remain in the cartel. This is an example of a tit-for-tat iterated prisoner’s dilemma, which can also be represented by a tree diagram as seen below.

Figure 7:

Example 3: Firm B is debating whether to enter an industry controlled by a monopoly. Firm A could either maintain current levels of output (allowing Firm B to enter), or increase output by investing in expensive machinery, a barrier to entry which would harm Firm B if it decides to enter. The payoff table shows all possible additions/reductions to profit below:

Figure 8:

A

Increase (1-p) Same (p)

B Enter $80m, -$50m $40m, $40m

Stay out $100m, $0 $50m, $0

p is the probability that Firm A maintains its level of output (because it won’t have access to machinery).

1-p is the probability that Firm A increases its output (as it has the capability to invest in machinery)

When probability is included, it is easier to represent the possible outcomes in a tree diagram.

Figure 9:

In this scenario, if B were to enter the industry when A increases its output by investing in machinery, A would decrease its prices to ease out the competition - eventually, B would lose profits – A would receive $80 million while B would lose $50 million. If Firm B was to enter the industry but A is unable to invest in machinery, the industry would become a duopoly and profits would (theoretically) be spread evenly between the two – each would receive $40 million. However, if B stayed out and A increased its output, A would be more productive and wouldn’t face competition, so it would gain the most profits at this point – A would receive $100 million while B receives $0. If Firm A stays the same and Firm B stays out, Firm A would continue to earn profits from the lack of competition but not as much due to x-inefficiency – A receives $50 million while B receives $0.

In this scenario, A would be better off increasing its machinery – its lowest payoff is $80m if it increases in size and $40m if it stays the same. Also, the highest payoff is $100m if it increases in size and $50m if it stays the same – there is no incentive not to stay the same size. Furthermore, whether B chooses to enter or stay out, the best possible outcome in each case occurs when A increases in size. Theoretically, firm B would be better off staying out of the industry as the worst possible payoff if it entered was -$50m, and $0m if it chose to stay out. Therefore, the Nash Equilibrium would be at (100, 0).

However, this does not take into account diseconomies of scale (disadvantages with increasing in size) and the probability (p) that Firm A won’t be able to increase its output by investing in machinery – if A won’t be able to do so, the Nash equilibrium will shift to (40, 40). The value of p could affect whether it would be rational for B to enter or stay out.

P can be found by creating a general formula for the payoff Firm B receives in each scenario. This is done by multiplying the probability with the payoff values, as seen in Figure 9.

If B stays out, the payoff B receives would be 0, as there is no production.

If B enters the market, the general formula for the payoff it would receive would be:

payoff=40p+-50(1-p)

payoff=90p-50

The payoff should be greater than 0; otherwise there is no incentive for B to enter the market.

90p-50>0

90p>50

p>5/9

Therefore, B will enter the market if the probability that A cannot invest in machinery and will have to keep their output levels the same is higher than 5/9. Firm B can also calculate its possible payoffs with a known value of p.

Example 3 can be represented by a general formula where u > y > w > x > z > v.

Figure 10:

A

Increase (1-p) Same (p)

B Enter u, v w, w

Stay out x, z y, z

A general formula for calculating p can be found where symbols used come from Figure 9:

payoff=w(p)+v(1-p)

payoff=(w-v)p+v

(w-v)p+v>z

(w-v)p>z-v

p>(z-v)/(w-v)

Hawk-Dove

Examples 4 and 5 are instances of hawk-dove games. A Hawk-Dove game, also known as Chicken, occurs when two players compete for a good of a known value (v) and there are two possible options “Hawk” or “Dove”. “Hawk” is considered to be the stronger, riskier strategy while “Dove” is considered to be the safe strategy. Players choose simultaneously. This originated as a biological game, but can be applied to Economics as it is used to model scenarios involving resources.

Example 4: Firm A is choosing to invest in technology. Its competitor, Firm B, is also considering doing so. Firm A is the same size as Firm B. The payoff table below shows the resulting profits for all possible combinations:

Figure 11:

A

Invest Don’t Invest

B Invest 20, 20 0, 50

Don’t Invest 50, 0 25, 25

In the example above, the profit from technology for both A and B is $50 million, the cost to investing is $10 million and Firm A is equal in size to Firm B (so the resulting payoffs are the same). The most optimal output appears to be $25 million for both, when both don’t invest in technology. However, the Nash Equilibrium, and most rational strategy, is for both firms to invest in technology because the lowest possible yield if they choose to invest is $20 Million, while the lowest possible yield if they choose not to invest is $0.

Example 5: Firm A choosing to invest in technology. Its competitor, Firm B, is also considering doing so. Firm A is double the size of Firm B. The payoff table below shows the resulting profits for all possible combinations:

This can also apply to different sized firms. If firm A is twice the size of firm B, the profit from investing in technology for firm A is $100 million, the profit from investing in technology for firm B is $50 million and the cost for investing is $10 million. The resulting payoffs are displayed in Figure 4 below. Again, though the most optimal decision appears to be when both firms don’t invest in technology, both firms have incentives to invest – for firm A, the lowest possible payoff is $45 million if it chooses to invest, and $0 if it doesn’t choose to invest, while for firm B, the lowest possible payoff is $20 million if it chooses to invest and $0 if it doesn’t. Therefore, Nash Equilibrium is at (45, 2) when both firms invest.

The general formula for Examples 4 and 5 is found below:

V is the value of the resource, while C is the cost incurred from fighting to obtain the resource. If the resource is shared between two, the value of it is halved, but when they end up fighting for it, they each incur a cost of C/2. When both choose hawk, each has a probability of winning by ½. The game is considered a type of prisoner’s dilemma when V>C. Though the most optimal decision appears to be when both firms choose the weaker, less aggressive option (Dove), Game theory shows that both firms should choose the more aggressive option (Hawk), even though it would incur costs as the payoff is higher than 0. However, when the cost incurred is higher than the value received (V

Zero-sum game

According to Investopedia, a zero-sum game is a situation in which a player’s gain is equal to another player’s loss, so the net change in benefit for both players is zero. It is a non-cooperative game. An example in economics is the futures market, while examples in other fields are poker and chess.

Example 6: Firm A is choosing to locate at a beach – either at the left, middle or right side. 60 customers are evenly spaced out across the beach. The payoff table shows the number of customers that each firm would have for each outcome.

Though having two firms selling the same product right next to each other may seem counterintuitive and a waste of resources, the Nash Equilibrium is achieved when both firms choose to locate middle. The highest possible payoff for both firms when they choose to locate in the middle is $40m, while the lowest possible payoff is $30m. When they choose to locate in either left or right, the highest possible payoff is $30m, and the lowest possible is $20m. Even though (30,30) can also be seen in 4 other instances (LL, LR, RL, RR), these aren’t equilibrium positions as there is still an incentive to betray the other by choosing the middle to earn $40m.

It can be deduced that there are 9 possible combinations from the table above, and by using combinations.

Survey

I carried out a survey using scenarios 1, 4, 5 and 6 from a sample of 48 Year 12 students (24 of which were Firm A and the other 24 were Firm B) who have never done any sort of Game Theory before to investigate how closely real-life data fits in with what is theoretically correct and rational (the theoretically correct option for each scenario is highlighted in Figures 14 and 15 below). I chose to do this because I was fascinated by how much the results of a simulation of Prisoner’s Dilemma my Economics teacher ran during one Economics class deviated from the rational. The questions on the survey (for Firm A) are reproduced below:

In all of the following scenarios, you are Firm A and Firm B is your competitor.

1: Prisoner’s Dilemma

You and a friend are arrested for a crime and sentenced to 2 years. Both of you are also suspected for a much larger crime (which you didn’t do), but the police do not have enough evidence to prove it. You are in solitary confinement with no means of speaking to the other. They give you a bargain – you could confess to committing the larger crime, betraying your friend, or deny it. The payoff table below shows the number of years you will receive for each possible outcome:

A

Confess Deny

B Confess 5, 5 10, 1

Deny 1, 10 2, 2

What would you do: Confess Deny

2: Tech

You are choosing to invest in technology. Your competitor is also considering doing so. You are the same size as Firm B. The payoff table below shows the addition/reduction to your current profits:

Would you invest in technology? Yes No

3: Tech

You are choosing to invest in technology. Your competitor is also considering doing so. You are twice the size of Firm B. The payoff table below shows the additional/reduction to your current profits:

Would you invest in technology? Yes No

4: Location

You are choosing to locate at a beach – either at the left, middle or right side. The customers are evenly spaced out across the beach. The payoff table shows the number of customers that each firm would have for each outcome.

Where would you locate? Left Middle Right

The results from the survey I carried out for the 4 scenarios above can be seen below:

Prior to the analysis carried out earlier (so if each subject randomly guessed an option) the options in scenarios 1, 4 and 5 would each have probabilities of 0.5, while options in scenario 6 would each have probabilities of 0.33. However, data from Figure 15 doesn’t reflect this.

There are 24 possible combinations each subject could take when answering the survey.

possible combinations = (_1

2)C × (_1

2)C × (_1

2)C × (_1

3)C

= 2 ×2 × 2 × 3

= 24

If each subject randomly guessed for the 4 scenarios, each combination would have a probability of 1/24 or 0.042, and a frequency of 2. However, data from Figure 16 also does not reflect this.

Taking into account the analysis done on the situations above, theory says that the rational decisions (highlighted) should have probabilities of 1 and total frequencies of 48, while the other options would have probabilities and frequencies of 0. Therefore, theory also says that the combination confess-invest-invest-middle would have a probability of 1 and a frequency of 48, while the other options wouldn’t.

It is evident from the data above in both figures that this is not the case in the real world. Reasons for deviating from the expected result included trusting the opponent (even without communication), being misled by the highest possible payoff (applicable to scenarios 1, 4 and 5) etc. Results for Firm A in scenario 5 deviated the most from the expected result (by 0.5), perhaps because people were misled by the change in size compared to the opposing firm.

Conclusion

Overall, in this exploration, I investigated three forms of Game Theory (Prisoner’s Dilemma, Hawk-Dove, Zero-sum) by looking at 6 scenarios: the basic prisoner’s dilemma, cartels, entering a monopolistic market, investing in technology depending on firm size and determining where to locate (left, middle or right) through modeling by constructing payoff tables and tree diagrams, then arriving at a general formula for each scenario. Some assumptions made include firm size, payoff values, number of options available, ceteris paribus (other factors unchanged), and constant payoff values (they wouldn’t change in the long run).

Through conducting a survey to obtain real-life data and comparing it to theory, I discovered that Game Theory has many limitations. The biggest underlying assumption behind Game Theory is that humans are rational – players are rational and know that their opponents are also rational. In reality, humans are very unpredictable and irrational, so real life data will almost always vary from theoretical results. The degree of uncertainty, a key element to all of these scenarios, cannot be predicted in real-life – some may be more trusting than others and therefore it would impact their decision-making. Moreover, in some scenarios, there is perfect knowledge. Other limitations are that humans may be swayed be the possibility of greater rewards and forego rationality or may carry the ‘high-risk, high win’ mentality. As a result, game theory, a form of prediction of human behavior in conflict scenarios, may be purely theoretical and may not reflect real-life. Also, there may have been errors in obtaining real-life data during the survey – subjects may have been lying about not knowing Game Theory or may not have completely understood the scenario or the payoff table.

However, in situations where decisions need to be made quickly without perfect knowledge, game theory is valuable in determining the best option possible, as rational decision-making is crucial in such scenarios. Outside Economics, Game Theory was particularly useful during the Cold War in the 1950s between USA and Russia, where governments discovered using game theory that it would be beneficial not to launch a nuclear bomb.