Deliverable 7 - Presentation on Game Theory Applications

Optimum Strategy of the game is (Sue, Sue) that is (5, -5)

Although selecting “Don’t Sue” will produce the highest potential payout to the competitor, since the manufacturer will be sticking to their dominant strategy of “Sue” they would actually be left with a loss of $20 million in annual profits going with that choice. Therefore, the best play the competitor can make is to choose to countersue the manufacturer since it will result in only $5 million lost in annual profits compared to the $20 million they’d lose by going with “Don’t Sue”. This is additionally supported by (Sue, Sue) being the only Nash Equilibrium point in the payoff matrix, meaning it is the strategic choice for both to make because neither will have the incentive to move.

6

If Results Match with those of My Co-workers:

Part 2

The results I got from the payoff matrix, do match with those of my co-workers

Because, the dominant strategy for the Manufacturer is to Sue

Based on the information I found from the payoff matrix I do agree with my coworker. She was correct that the dominant strategy for the manufacturer is “Sue” since those two values when the competitor chooses to “sue” or “don’t sue” are their highest possible payouts. The competitor does not have a dominant strategy since they select to “Sue” if the manufacturer sues and chooses “Don’t Sue” when the manufacturer also chooses to not sue. Although selecting “Don’t Sue” will produce the highest potential payout to the competitor, since the manufacturer will be sticking to their dominant strategy of “Sue” they would actually be left with a loss of $20 million in annual profits going with that choice. Therefore, the best play the competitor can make is to choose to countersue the manufacturer since it will result in only $5 million lost in annual profits compared to the $20 million they’d lose by going with “Don’t Sue”. This is additionally supported by (Sue, Sue) being the only Nash Equilibrium point in the payoff matrix, meaning it is the strategic choice for both to make because neither will have the incentive to move.

7

Using Mixed Strategy Algorithm to Find Optimum Manufacturer’s Strategy:

Part 3

= -5p + 20(1-p) = -5p + 20 – 20p = 15p + 20

= 10p + 15(1-p) = 10p + 15 – 15p = 25p +15

15p + 20 = 25p +15

(-15p) = (-15p)

20 = 10p +15

(-15) = (-15)

5 = 10p

𝑃 = 5/10 = 1/2 or .50 or 50%

In order to make sure my coworker’s optimum strategies for the manufacturer and the competitor are correct, I decided to use the same payoff matrix payouts and worked through finding the optimum strategy for both the manufacturer and the competitor using a mixed strategy algorithm. The use of the mixed strategy algorithm is due to there being no present dominant strategies for either player in the nonzero-sum game.

First I found the mixed strategy algorithm for the manufacturer. To find the optimum strategy for the manufacturer I first needed to set up the expected values for the competitor’s two choices where they will be equal in their payouts, making the competitor indifferent to the manufacturer’s chosen strategy. When setting up the expected values, “Sue” will be represented as 𝐸_𝑆 and “Don’t Sue” to be represented as 𝐸_𝐷 for their expected values. What we aim for is 𝐸_𝑆 = 𝐸_𝐷.

As shown in the break down of how I found my expected values for the competitor’s “Sue” column, expected value comes out to be 𝐸_𝑆=15p + 20 where -5 = 𝑃 and 20 = 1-𝑃. For the expected value for the “Don’t Sue” column, 𝐸_𝐷= 25p + 15, where 10 = 𝑃 and 15 = 1-𝑃. By simplifying and balancing out the two expected values I was able to find that the optimum strategy for the manufacturer is to choose “Sue” 50% of the time, and “Don’t Sue” the other 50% of the time.

8

Using Mixed Strategy Algorithm to Find Optimum Competitor’s Strategy:

Part 3

𝐸𝑆= 5p + 20(1-p) = 5p + 20 – 20p = 15p + 20

𝐸𝐷= 10p + 15(1-p) = 10p + 15 -15p = 25p +15

15p + 20 = 25p +15

(-15p) = (-15p)

20 = 10p + 15

(-15) = (-15)

5 = 10p

𝑷 = 5/10 or 1/2 or .50 or 50%

Manufacturer Optimal Strategy:

Choose to sue 50% (1/2) of the time

Choose to not sue 50% (1/2) of the time

Competitor Optimal Strategy:

Choose to sue 50% (1/2) of the time

Choose to not sue 50% (1/2) of the time

Next, I needed to find the optimum strategy for the competitor, to make sure that my coworker was correct in their work stating that their optimum strategy is the same as the manufacturer’s. To find the optimum strategy for the competitor I first needed to set up the expected values for the manufacturer’s two choices where they will be equal in their payouts, making the manufacturer indifferent to the competitor’s chosen strategy. When setting up the expected values, “Sue” will be represented as 𝐸_𝑆 and “Don’t Sue” to be represented as 𝐸_𝐷 for their expected values. What we aim for is 𝐸_𝑆 = 𝐸_𝐷.

As shown in the break down of how I found my expected values for the manufacturer’s “Sue” row, expected value comes out to be 𝐸_𝑆=15p + 20 where 5 = 𝑃 and 20 = 1-𝑃. For the expected value for the “Don’t Sue” row, 𝐸_𝐷= 25p + 15, where 10 = 𝑃 and 15 = 1-𝑃. By simplifying and balancing out the two expected values I was able to find that the optimum strategy for the competitor is the same as the manufacturer’s; to choose “Sue” 50% of the time, and “Don’t Sue” the other 50% of the time.

9

If Results Match those of My Co-workers:

Part 3

The results do Match those of my coworker’s conclusion based on the payoff matrix & my results using that data.

Optimum strategy for both the manufacturer & competitor is opt for “Sue” half the time, and “Don’t Sue” the other half of the time

Since our calculations turn out to be the same, this would mean that I did agree with my coworker’s conclusion that the optimum strategy for both the manufacturer and competitor is the choose “Sue” half the time, and “Don’t Sue” the other half of the time. There is one issue though, this solution is not realistic since neither company can select a strategy more than once. Both the manufacturer and the competitor get once chance to choose to either sue the other or not to. This then means that the optimal strategy shows which company has the bigger advantage or best shot at winning a suit. In this case, both are split 50/50 in their decision, so neither has the upper hand in the decision they make.

10

Game Tree for this Scenario With Any Non-Credible Threats Excluded & the First Step of Backwards Induction

Part 4

Manufacturer

Sue

Competitor

Sue

(5, -5)

(20, -10)

Don’t Sue

Competitor

Don’t Sue

(15, 15)

Don’t Sue

Manufacturer

Sue

Don’t Sue

Competitor

Competitor

Sue

Don’t Sue

(5, -5)

(15, 15)

Game Tree With Any Non-Credible Threats Excluded

First Step of Backwards Induction

Once all branches had been established on the game tree, their corresponding payouts were listed below each of the four possible strategies. From here you would begin to perform backward induction to find the optimal strategy, however, within our game tree there is a non-credible threat that must first be removed. The non-credible threat that needs to be removed as it doesn’t make sense in our scenario would be the (Don’t Sue, Sue) branch. This needs to be removed because if the manufacturer were to decide to not sue the competitor they wouldn’t be left with a choice to countersue them because the game would be complete at that point with the manufacturer having made their decision to not sue making the competitors only choice as to also not sue. Once that branch has been removed, we can then continue on to the backward induction to find the optimal strategy.

11

Second Step of backwards Induction:

Part 4

Manufacturer

Sue

Competitor

(5, -5)

Competitor

Don’t Sue

(15, 15)

Then, in order to find the optimal strategy we must remove the weaker choice(s) branch. Leaving us with just the top portion of the game tree (Sue, Sue) and (Don’t Sue, Don’t Sue).

12

Optimum Strategy For the Game:

Part 4

Manufacturer

Sue

Competitor

(5, -5)

Competitor

Don’t Sue

(15, 15)

Optimal Strategy for the game is: (Don’t Sue, Don’t Sue), that is, (15, 15)