Question

Game of Games Q18-Q20 concern the following game Erving and Dorothy are playing the popular online roleplaying game World of Warcraft. Players advance in the game by accumulating experience points. Erving and Dorothy have heard that a powerful monster, the Elder Bruin, will appear if you do an Eight Clap in the center of Westwood Forest. If they fight the Elder Bruin together, they will win and will each get 10 experience points (a payoff of 10 each). After a little preparation, they walk together to the center of the forest, and do the Eight Clap. But in the seconds before the Elder Bruin appears, both notice that many weaker monsters are wandering around nearby. Erving and Dorothy realize that they could just run away and fight those monsters instead of fighting the approaching Elder Bruin. With seconds to choose, each must independently decide to stay and fight the Elder Bruin or run off to fight the weaker monsters. If Erving sticks to the plan but Dorothy runs off to fight the weaker monsters, Erving will be unable to beat the Elder Bruin alone and will receive 0 experience points. Dorothy, by contrast, will receive 6 experience points for killing the weaker monsters. The same is true if Dorothy sticks to the plan and Erving doesn't: 0 for Dorothy and 6 for Erving. If they both run away to fight the weaker monsters, the Elder Bruin won't chase them, and they will each get 6 experience points for killing the weak monsters. To summarize: if both fight the Elder Bruin, both get a payoff of 10; if one fights the Elder Bruin but the other runs away to fight the weak monsters, the one fighting the Elder Bruin gets a payoff of O and the one fighting weak monsters gets a payoff of 6; if both fight the weak monsters, both get a payoff of 6. Assume both players only care about experience points. Q18. Draw the payoff matrix for this game. (5 points) 19. Does this game have any pure-strategy Nash equilibria? If so, find them. Explain your answer by checking all possible outcomes. (S points) 020. Which outcome is socially optimal? Briefly explain how you determined this. (2 points)

Answer 1

Let’s represent Dorothy as D and Erving as E.

There are two actions either to fight Elder Bruin (Represent it by F) or run off to fight Weak Monsters (Represent it by R)

If both D & E plays F payoff will be 10 for both

If both D & E plays R payoff will be 6 for both

If D plays F and E plays R payoff for D will be 0 and for E will be 6

If D plays R and E plays F payoff for D will be 6 and for E will be 0

1. Let Erving be Column Player and Dorothy be Row Player Payoff Matrix will be as follows:

		Player Erving (E)
		Fight Elder Bruin (F)	Runoff to fight Weak Monsters (R)
Player Dorothy (D)	Fight Elder Bruin (F)	10,10	0,6
	Runoff to fight Weak Monsters (R)	6,0	6,6

2. Finding Nash Equilibria

Let’s analyse the response of player D. If E play F the possible payoff for D will be 10 for playing F and 6 for playing R, hence D will play F as payoff is higher. Similarly, if E play R the possible payoff for D will be 0 for playing F and 6 for playing R, hence D will play R as payoff is higher. Besr Responses of D are shown in payoff matrix as Red Circle

Player Erving (E) Fight Elder Bruin (F) Runoff to fight Weak Monsters (R) Player Dorothy (D Fight Elder Bruin (F) 0,6 Runoff to fight Weak Monsters (R) 6 6,0

Now, let’s analyse the response of player E. If D play F the possible payoff for E will be 10 for playing F and 6 for playing R, hence E will play F as payoff is higher. Similarly, if D play R the possible payoff for E will be 0 for playing F and 6 for playing R, hence E will play R as payoff is higher. Best Responses of E are shown in payoff matrix as Blue Square.

Player Erving (E) Fight Elder Bruin (F) Runoff to fight Weak Monsters (R) Player Dorothy (D)Fight Elder Bruin (F) 0,6 Runoff to fight Weak Monsters (R) 6 6,0

There are two outcome were Best Responses intersect. Further, either player has no benefit by deviating to other action. Consider F,F outcome, if D or E shifts from F to R it will reduce the payoff from 10 to 6. Consider R,R outcome, if D or E shifts from R to F it will reduce the payoff from 6 to 0. Hence, there are two pure strategy Nash Equilibria i.e F,F and R,R. So, either both Erving and Dorothy will fight Elder Bruin or runoff to fight Weak monsters.

C. Socially Optimal Outcome

Socially Optimal Outcome is F,F as it gives the maximum payoff to both the players. Shifting from F,F to any other Outcome will not increase the payoff of both the players simultaneously. Shifting from F,F to R,R will reduce the payoff from 10,10 to 6,6. Shifting from F,F to R,F will reduce the payoff from 10,10 to 6,0. Shifting from F,F to F,R will reduce the payoff from 10,10 to 0,6. Hence F,F is the only socially optimal solution.