4. Computing mixed-strategies equilibria

Just like any other undergraduate student, Antonio knows the importance of arriving before his roommate when it comes to moving into his dorm room. However, he also recognizes the difficulty in moving at exactly the same time as his roommate; everyone is cramped in the stairways, hallways, and in the elevator, and there's a limit to the number of bodies that can fit in that small room at once to unpack.

Since Antonio is a first-year student, the university assigned Dmitri to be his roommate. Assume that it is impossible for either roommate to get in touch with the other one prior to move-in day. So each roommate must decide simultaneously whether to arrive in the morning or afternoon. If they move in at the same time, each receives a payoff of 0; if they move in at different times, each receives a positive payoff, but the person arriving in the morning is better off. The following payoff matrix represents the game they must play.

See table 1: DmitriMorningAfternoonAntonioMorning0,045,30Afternoon30,450,0

Amixed-strategy equilibriumis one in which each player assigns a probability distribution over the possible pure strategies. In this particular game, this means that each person will play Morning with a certain probability and Afternoon with 1 minus that probability. To solve for the mixed-strategy equilibria, you must first determine each person's best response, given the strategy of the other player.

Let? ?be the probability that Dmitri plays Morning, and? ?be the probability that Antonio plays Morning. Based on the payoff matrix, you know that each player's goal is to arrive on move-in day at different times.

Use the following table to match Antonio's best response when Dmitri plays?=1

?=1and?=0

See Table 2

It turns out that for almost all values of? , Antonio's best response is to play?=0

?=0or?=1 ?=1. However, there is a certain value of?that Dmitri may play where Antonio is actually indifferent between all possible mixed strategies that he could play (the proof of this is beyond the scope of this question). To determine this value for?, you can simply set the expected payoff (EP) for Antonio when he plays Morning with 100% probability (?=1

?=1) equal to the expected payoff when he plays Afternoon with 100% probability (?=0

?=0), given that Dmitri plays Morning with probability?:

Antonio's EP from Always Playing Morning=Antonio's EP from Always Playing Afternoon = ?=

Using the same logic, there exists a unique value for?where Dmitri is indifferent between playing Morning with 100% probability (?=1) and Afternoon with 100% probability (?=0), given that Antonio plays Morning with probability?:

Dmitri's EP from Always Playing Morning=Dmitri's EP from Always Playing Afternoon = ?=

On the following graph, use the blue points (circle symbol) to plot Antonio's best-response function (BRF), and use the purple points (diamond symbol) to plot Dmitri's best-response function. Line segments will automatically connect the points. Remember to plot from left to right. Then, place black points (plus symbol) to indicate all mixed strategy Nash equilibria of this game.

See Graph

4. Computing mixed-strategies equilibria Just like any other undergraduate student, Antonio knows the importance of arriving before his roommate when it comes to moving into his dorm room. However, he also recognizes the difficulty in moving at exactly the same time as his roommate; everyone is cramped In the stairways, hallways, and in the elevator, and there's a limit to the number of bodies that can fit in that small room at once to unpack. Since Antonio is a first-year student, the university assigned Dmitri to be his roommate. Assume that it is impossible for either roommate to get in touch with the other one prior to move-in day. 50 each roommate must decide simultaneously whether to arrive in the morning or afternoon. If they move in at the same time, each receives a payoff of 0; if they move in at different times. each receives a positive payoff. but the person arriving in the morning is better off. The following payoff matrix represents the game they must play- Dmitri Moming Anomoon DELL FI ES FO FV D Q W E R T Y U O A S D F G H J K ckDmitri Morning Afternoon Antonio Morning 0 0 45, 30 Altemoon 30, 45 0. 0 A mixed-strategy equilibrium is one in which each player assigns a probability distribution over the possible pure strategies. In this particular game, this means that each person will play Morning with a certain probability and Afternoon with 1 minus that probability. To solve for the mixed- strategy equilibria, you must first determine each person's best response, given the strategy of the other player. Let A be the probability that Dmitri plays Morning, and p be the probability that Antonio plays Morning. Based on the payoff matrix, you know that each player's goal is to arrive on move-in day at different times. Use the following table to match Antonio's best response when Dmitri plays A = 1 and A = 0. O DELL FI F9 Fa FO %Homework (Ch 05) Use the following table to match Antonio's best response when Dmitri plays A = 1 and A = 0. A=1 A=0 P=0 O O p=1 O O It turns out that for almost all values of A, Antonio's best response is to play p = 0 or p = 1. However, there is a certain value of A that Dmitri may play where Antonio is actually indifferent between all possible mixed strategies that he could play ( the proof of this is beyond the scope of this question). To determine this value for A, you can simply set the expected payoff (EP) for Antonio when he plays Morning with 100%% probabury [ pi= 1) equal to the expected payoff when he plays Afternoon with 100% probability (p = (]. given that Dmitri plays Morning with probability A Metonic's EP from Always Playing Morning Antonio's EP from Always Playing Afternoon omg Ba sara logic, there exists a benique value for p where Omnitel is indifferent between playing Morning with 100% probability IA - 1) and O 9 DELL 2 U OQ Search Homework (Ch 05) Antonio's EP from Always Playing Morning Antonio's EP from Always Playing Afternoon Using the same logic, there exists a unique value for p where Dmitri is indifferent between playing Morning with 100% probability (A =1) and Afternoon with 100% probability () = 0), given that Antonio plays Morning with probability p: Dmitri's EP from Always Playing Morning Dmitri's EP from Always Playing Afternoon On the following graph, use the blue points (circle symbol) to plot Antonio's best-response function (BRF), and use the purple point (diamond symbol) to plot Dmitri's best-response function. Line segments will automaticaly connect the points. Remember to plot from left to right Then, place black points (plus symbol) to indicate all mixed strategy Nash equilibria of this game. O 9 AD DOLL FS FO Fe