There are 2 drivers who must travel from point A to point B. Each driver prefers
ID: 1167639 • Letter: T
Question
There are 2 drivers who must travel from point A to point B. Each driver prefers a shorter travel time. Two routes are available, one via point X and one via point Y . On the A ? X path one car takes 10 minutes and two cars take 20 minutes each. On the X ? B path each car takes 25 minutes (no matter if there are one or two cars on this path). On the A ? Y path each car takes 25 minutes (no matter if there are one or two cars on this path). On the Y ? B path one car takes 10 minutes and two cars take 20 minutes each. This network is illustrated in Figure 1a.
a) Model this situation as a Strategic Game (defining players, action sets and payoff functions) and draw the payoff matrix of the game.
b) Find all Nash Equilibria.
c) Now suppose that a new road is built allowing any car to travel instantly from X to Y (but cannot be used to travel from Y to X). See Figure 1b. Draw the payoff matrix of the new game and find all Nash Equilibria. Has the new road reduced the average travel time?
d) Consider the following extension of the previous problem. There are four drivers. On the A ? X path one car takes 10 minutes and each additional car increases the travel time per car by 10 minutes. On the X ? B path each car takes 45 minutes (no matter how many cars are on this path). On the A ? Y path each car takes 45 minutes (no matter how may cars are on this path). On the Y ? B path one car takes 10 minutes and each additional car increases the travel time per car by 10 minutes. This network is illustrated in Figure 1c. Find all Nash Equilbria.
e) Now suppose that a new road is built allowing any car to travel instantly from X to Y (but cannot be used to travel from Y to X). See Figure 1d. Using your results from part c), guess a Nash Equilibrium of this game and prove that it is indeed a Nash Equilibrium.
10,20 25 10,20 25 10,20,30,40 45 10,20,30,40 45 B A B A 0 25 25 45 45 10,20 10,20 10,20,30,40 10,20,30,40 Figure 1Explanation / Answer
Ans)
a)Players: 2 people
Actions: choosing routes between A-X-B and A-Y-B
Pay-off: Negative of total travel time
Player 2 travels through X
Player 2 travels through Y
Player 1 travels through X
-45,-45
-35,-35
Player 1 travels through Y
-35,-35
-45,-45
b) If both players choose the path A-X-B, then time taken by each player = 20+25 = 45 minutes.
If both players choose the path A-Y-B, then time taken by each player = 20+25 = 45 minutes.
But in both the cases the players can increase their pay-offs and reduce their travel time, if any one player deviates it’s route i.e is 10+25=35 minutes.
Thus, here the Nash Equilibrium is that one driver will take the A-X-B route and the other will take A-Y-B route.
c)
Player 2 travels through X
Player 2 travels through Y
Player 2 travels through X and Y
Player 1 travels through X
-45,-45
-35,-35
-45,-20
Player 1 travels through Y
-35,-35
-45,-45
-45,-20
Player 1 travels through X and Y
-20, -45
-20, -45
-40, -40
If both players choose the path A-X-Y-B, then time taken by each player = 20+20 = 40 minutes.
If both player choose the path A-X-B or A-Y-B, then time taken by each player = 20+25 = 45 minutes.
If one player choose the path A-X-B and another the A-Y-B path, then time taken by each player = 10+25 = 35 minutes.
If one player choose the path A-X-Y-B, then time taken by the player = 10+10 = 20 minutes and the other driver takes the A-Y-B or A-X-B path, then time taken by the player = 20+25 = 45 minutes.
Thus, here the Nash Equilibrium is that If one player choose the path A-X-B and another the A-Y-B path, then time taken by each player = 10+25 = 35 minutes.Thus, the construction of the road has not reduced the average travel time.
d) Players: 4 people
Actions: choosing routes between A-X-B and A-Y-B
Pay-off: Negative of total travel time
If all 4 players choose the same path, either A-X-B or A-Y-B, then time taken by each player = 40+45 = 85 minutes.
If any one player deviates from above & choose the other path, then time taken by that player = 10+45 = 55 minutes, while the pay-off for other 3 o changes as, time taken is = 30+45=75.
But in both the cases the players can increase their pay-offs and reduce their travel time, if any 2 players take the A-X-B route and other 2 take the A-Y-B route as time take for each player is 20+45=65 minutes. Thus, this is the Nash Equilibrium.
e) Moving on similar lines to what we have seen in above cases, in this case the Nash Equilibrium will be that 2 players will take the A-X-B route with time required =20+45=65 minutes and 2 players will take any of the A-Y-B route with time required as 20+45=65 minutes. Any deviation from this, will reduce the player’s pay-off and increase the time taken.
Player 2 travels through X
Player 2 travels through Y
Player 1 travels through X
-45,-45
-35,-35
Player 1 travels through Y
-35,-35
-45,-45
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