Problem 1 (To Make Roshambo (Rochambeau) Interesting)

You made a $86 bet with your friend, Bob, over a game of Rock, Paper, Scissors (RPS, Figure 1). The bet is to play a single game of RPS, in which the loser pays the winner $86. In the event of a draw, you pay Bob $10. Your friend Bob thinks that you are at a disadvantage in this bet, but you secretly have a trick up your sleeve!

You have previously studied the game of RPS and identified that there are two types of RPS players, to which you assign a random variable T = 1 and T = 2, respectively (T for type). As a result of your dedicated study, you identified the trends of weapons T = 1 players prefer to play versus the trends of weapons T = 2 players prefer to play. We will denote the weapon played by Bob by the random variable W (it has 3 possible values: R, P and S.)

Figure 1: The chart of Rock Paper Scissors. By Enzoklop (CC BY-SA 3.0) (link to source)

In preparation for the big game, you build a probability model that reasons about the likelihood of playing each weapon, for each type of player T. This model is written in terms of the conditional probability P(W|T), and is summarized in Table 1

Table 1: Your Roshambo probability model

Finally, you define a random variable X to be your return as a result of the game: $86 if you win, −$10 if you draw and −$86 if you lose.

You don’t know which type of player Bob is, but would like to study his previous games in order to identify his type; all in an effort to maximize your expected return.

(a) Let’s investigate first what happens if you play Rock in the big game. To start, we will assume, for now, that Bob is a T = 1 player. Calculate the distribution of X|T = 1 assuming you play Rock. That is, fill the values in Table 2.

Table 2: Distribution of return if you play Rock against a T = 1 player; Problem 1 Part (a)

(b) We’re still investigating what happens if you play Rock in the big game. This time, we will assume (for now) that Bob is a T = 2 player. Calculate the distribution of X|T = 2 assuming you play Rock. That is, fill the values in Table 3.

Table 3: Distribution of return if you play Rock against a T = 2 player; Problem 1 Part (b)

(c) We will now investigate what happens if you play Paper in the big game. Calculate the distributions of X|T = t assuming you play Paper, for t = 1,2. That is, fill the values in Table 4.

Table 4: Distributions of return if you play Paper against the different types of players; Problem 1 Part (c)

(d) We will now investigate what happens if you play Scissors in the big game. Calculate the distributions of X|T = t assuming you play Scissors, for t = 1,2. That is, fill the values in Table 5.

Table 5: Distributions of return if you play Scissors against the different types of players; Problem 1 Part (d)

We now get to the issue of not knowing what type of player Bob is. To complicate things further, you got inside information that Bob actually rotates between the different types T = 1,2. This information says that with probability θ, Bob selects to be player type T = 1 in any given game; and with probability 1−θ (the remaining probability), Bob selects to be player type T = 2 in any given game. Bobs various selections of the type of player he would like to be in each game are independent of each other. We can also assume that the weapons Bob selects each game are independent of each other.

In order to estiamte the value of θ, we rely on maximum likelihood and the data from Bob’s previous games.

(e) Before we get to the data, we first want to calculate the likelihood (as a function of θ) that Bob plays each possible weapon W. Calculate the probability P(W;θ) (as a function of θ). In other words, fill the formulas in Table 6

Table 6: Distributions of W; Problem 1 Part (e)

(f) Assume we have data from n games that Bob previously played. In this dataset D, we observe the weapon Wi played by Bob in each game i = 1,2,...,n. Concretely, D = {W1,...,Wn}. Assume that in nr games, Bob played Rock; in np games, Bob played Paper; and in ns games, Bob played Scissors. Write a formula for the likelihood of the whole dataset, as a function of θ,nr ,np and ns , P(D;θ,nr ,np,ns).

(g) Calculate the log-likelihood of the likelihood calculated in the previous part.

(h) We now look at the data and figure out that out of 3529 games Bob played, nr = 1145,np = 1283 and ns = 1101. Use the log-likelihood to find θˆ, the maximum likelihood estimate of θ, given this data.

(i) For this part you may use the value θˆ = 0.87. Calculate your expected return for each possible weapon you can play in the big game, using the estimate that Bob is a T = 1 player with probability θˆ and a T = 2 player with probability 1−θˆ. In other words, calculate E[X;θˆ = 0.87] assuming you play 1) Rock, 2) Paper, and 3) Scissors and fill Table 7. Given these values, if you had to choose a deterministic strategy (i.e., pre-determine your move) against Bob, what weapon would you play in the big game in order to maximize your expected return from the bet?

Was this a good bet for you after all?

Table 7: Expected returns for you against Bob assuming θˆ = 0.87; Problem 1 Part (i)