Question

Think of a time when you were involved in strategic decision making. This could be a business situation or a personal situation. It could be anything from purchasing inputs for a manufacturing firm to trying to divide up household chores. Strategy is huge in sports – Should we punt or go for it on 4^th?

When I purchased my car. I looked for the type and color of the car first in the city and online. I looked for a certain price range and distance. I then contacted my credit union to see exactly how much I qualified for and then I began the more serious process by test driving cars here in the city and speaking with salesman over the phone from online cites out of the city. Of course let me not forget I also looked for the cars with miles under 50,000.

Discuss questions

1.Discuss your dominant strategy for the situation.

2.What was the other person’s dominant strategy?

3.Was the outcome a Nash equilibrium? Why or why not?

Please answer each part of the questions and use game theory terminology. Make sure that you have an adversarial situation with at least two parties.

Have fun with this! These are useful tools to understand for your everyday life!

Game Theory Help!

For questions 1-3, you will need to determine whether or not each firm has a dominant strategy in order to know if the game is a Prisoners’ Dilemma.

A dominant strategy is one in which the agent (the firm, in this case) would have no incentive to change its strategy regardless of what the other agent does. I like to do this...

1. Pretend that Lowe’s chooses first. Write down Lowe's best response to each of Home Depot's choices. That is, what should Lowe’s do if Home Depot picks “cooperate” and what should Lowe’s do if Home Depot picks “don’t cooperate”.

2. Do the same for Home Depot choosing its strategy first. Write down Lowe's best response.

3. If you get that the firm always chooses “cooperate” for example, then “cooperate” is the dominant strategy. If the firm always chooses “don’t cooperate”, then “don’t cooperate” is its dominant strategy. If you get one of each chosen, there is no dominant strategy for that firm.

Questions 4-7 are very similar, but you will need to determine the Nash and strategically stable outcome (if there is one)

1. Pretend that the Saint Petersburg Times chooses first. Write down Tampa's best response to each of SPT's choices. That is, what should Tampa do if SPT picks low price and what should Tampa do if SPT picks high.

2. Do the same for Tampa choosing its price first. Write down SPT's best response.

3. If you get that the firm always chooses Low, for example, then Low is the dominant strategy. If the firm always chooses High, then High is its dominant strategy. If you get one of each chosen, there is no dominant strategy for that firm.

4. To get the Nash Equilibrium, for each firm, ask yourself, "What is the worst case scenario for this firm." Then, pick the opposite strategy. Put a little dot in that square (the opposite of the worst case scenario). Do the same for the other firm. If there is a Nash, it will be the square with 2 dots.

5. Finally, the Nash equilibrium is the only strategically stable outcome. Both firms are doing the best that they can, so there is no incentive for them to deviate from that strategy.

Answer 1

Part-A

1) Suppose we assume a simple example of a game with two players.There are payoffs for each player as the outcome of the game.Now let us assume that in total we have four outcomes(two for player A and two for player B).Now both players can either go two ways(left or right)

We can further assume the outcome as (A,B):

If both players choose Left:outcomes are (1,1)

If Player A chooses left and Player B chooses right:outcomes are (0,1)

If both players choose right:outcomes are (0,0)

If plaer A chooses right and player B chooses left:outcomes are (0,0)

Now notice that if player B chooses right the possible payoffs for player are both 0 if player A chooses left or right.Hence,average pay off of player A if player B chooses right is also 0(0+0/2).

On the other hand if player B chooses left the payoffs of player A are 1 if player A chooses left and 0 if player B chooses right.Therefore, average payoff by player A if player B chooses left=(1+0)/2=0.5.

Therefore, player A is better off chosing left no matter what is chosen by player B as in case the average payoff from choosing left is higher than that of choosing right for player A regardless of what option is chosen by player B.Thus, choosing left would be the dominant strategy of player A.

2) Now, if player B chooses right regardless of what is chosen by player A the payoffs of player B are 1 and 0.Therefore, average payoff for player B from choosing right=(1+0)/2=0.5

If player B decides to chose left regardless of what is chosen by player A its payoffs are also 1 and 0 making the average payoff as (1+0)/2=0.5

Hence,player B would be indifferent between choosing left or right as the average payoffs are same for both the options.Thus,for player B in this case there is no dominant strategy unlike player A.

3) Nash Equilibirum refers to an equilibrium outcome of the game where all the players are able to achieve the best optimal outcome based on their respective desires.Now,in this particular case,only player A has a dominant strategy and player B does not there is no unique Nash Equilibria.Assunimg that the game is repeated multiple times and both players are not aware of each other's outcome priorly then player A will continue to choose left as the dominant strategy which will be its nash equilibria but player B will not have any nash equilibria if the conditions of the game remain the same all the time and hence,there will no unique Nash Equilbrium situation.

Part-B

1) Again,we assume that there are two situations of the game involving Lowe's and Home Depot.One is "Cooperate" and another is "Don't Cooperate".The payoffs or profits of Lowe's and Home Depot are given as (L,H)

Let's assume that if both chooses to Cooperate:the outcomes are (2000,3000)

If Lowe's chooses to cooperate and Home Depot doesn't:outcomes are (0,10000)

If both chooses don't cooperate:outcomes are (5000,150000)

If Lowe's chooses don't cooperate and Home Depot chooses to cooperate:outcomes are (10000,5000)

Now,notice that the payoffs for Lowe's for choosing to cooperate regardless of what is chosen by Home Depot are 2000 and 0.Hence,average payoff for Lowe's from choosing to cooperate=(2000+0)/2=1000

If Lowe's chooses don't cooperate then its outcomes are 5000 and 10000 making the average payoff as=(10000+5000)/2=7500

Therefore, clearly the average payoff is much higher from choosing not to cooperate for Lowe's making it the dominant strategy of the firm regardless of what strategy is chosen by its competitor Home Depot.

2) Similarly, for Home Depot the payoffs from choosing to cooperate are 3000 and 5000 regardless of Lowe's strategy.Average payoff for Home Depot from choosing to cooperate=(3000+5000)/2=4000

If Home Depot decides don't cooperate then its profits or payoffs are 10000 and 15000.Hence,average payoff for Home Depot for choosing to not cooperate=(10000+15000)/2=12500

Again,clearly the average payoff from choosing don't cooperate is signficantly higher than choosing to cooperate for Home Depot.Thus,the dominant strategy for Home Depot would be to not cooperate regardless of Lowe's strategy.

3) In this scenario both the firms have a dominant strategy implying that the game does have a unique Nash Equilibrium.Here the dominant strategy of Lowe's and Home Depot is "don't cooperate".Therefore, the unique Nash Equilibrium of this game is that both firms will not cooperate atleast presumably given the expected profits or payoffs.

Part-C

1) Two players are involved in this game:St.Petersburg Times(SPT) and Tampa

Players can choose either high price and low price.

Let's assume that the possible payoffs for SPT and Tampa are given as (SPT,Tampa)

Suppose, that both chooses low price:outcomes are (3,4)

If SPT chooses low price and Tampa high price:outcomes are (0,0)

If both chooses high price:outcomes are (4.3)

If SPT chooses high price and Tampa low price:outcomes are (0,0)

Here,SPT's payoff from choosing low price are 3(If Tampa chooses low price as well) and 0(if Tampa chooses high price).Thus average payoff for SPT from choosing low price=(3+0)/2=1.5

Alternatively,SPT's payoffs from choosing high price are 4(if Tampa also chooses high price) and 0(if Tampa chooses low price).Average payoff for SPT from choosing high price=(4+0)/2=2

Best Response for SPT if Tampa chooses low price is to choose low price as well and high price if Tampa chooses high price.

SPT's average payoff from choosing high price is higher than that of choosing low price.Hence, the dominant strategy for SPT in this case is choosing high price.

2) If Tampa chooses low price its payoffs are 4(If SPT also chooses low price) and 0(if SPT chooses high price).Hence,average payoff for Tampa from choosing low price=(4+0)/2=2

The payoffs for Tampa from choosing high price are 3(if SPT chooses high price) and 0(if SPT prefers low price).Average payoff for Tampa from choosing high price=(3+0)/2=1.5

Best Response for Tampa if SPT chooses low price is to choose low price as well and if SPT chooses high price best response of Tampa is to high price.

Hence, the dominant strategy for Tampa is to choose low price comparing the average payoffs between choosing low and high price.

3) From questions 1) and 2) the dominant strategies for SPT and Tampa are (high price,low price).Hence,in this case both firms have a dominant strategy.

4) Based on the strategy chosen by the rival firm and if the game is repeated multiple times and the conditions of the game remain the same,both SPT and Tampa would presumably continue to stick to their dominant strategy thereby generating a unique outcome of the repeated games.Therefore, each players' dominant strategy is a function of all the strategies of the rival player.

5) As both firms have a dominant strategy,we have a unique Nash Equilibrium in this game.