Question

Joanna and Joe both love pizza. They go to a famous pizza bar in mid-town Manhattan called Sloppy Pizza. When they reach there, they hear about a contest that’s going on. They want to share the “Marvellous Pizza” together. That’s an8 slice pizza which is amazing!

The owner tells them to write on a piece of paper how many slices they each want out of the 8 slices of pizza. He puts them on separate tables and makes sure they don’t see each other’s chosen number. Thus the choices are made simultaneously. Each player can ask for any number of slices between 1 and 8 both inclusive, integer numbers only.

If the sum of the numbers they each write is less than or equal to 8, they each get the number of slices they named for free! If the sum of the numbers they each write is greater than8, and their chosen numbers are different, then the one with the smaller number gets the pizza slices theywrote (for free) and the other player is asked to leave. If sum of the numbers they each write is greater than 8, and they choose equal slices, they both get 4 slices each. Each player’s utility is the number of pizza slices he gets to eat.

1. Model the game in strategic form
2. Solve for all pure strategy Nash

Answer 1

a. In this case, there are two players in the concerned game and each player can choose any number of slices of the pizza between 1 and 8 including both these numbers or integers. If the sum of the number of pizza slices chosen by both players is less than or equal to 8 then both the players would get their preferred number of slices for free and if it is greater than 8 and the number of slices chosen by both players are different, then the player choosing the lesser number of slices will get them for free and the other player would have to leave. Now, if the sum of the slices chosen by both players is greater than 8 and both players end up choosing the same or identical number of pizza slices then both players would get 4 slices of pizza each for free. Therefore, both players in this game has 8 numbers or integers to choose from and the ultimate or subsequent payoff of both players ideally depends on the number of slices chosen by the other player or the competitor in the game. Alternatively, it can be considered that each player in this game has 8 individual strategies that they can possibly choose from and depending on te other player's strategy, the subsequent payoff or utility of each player would be determined which is equal to the number of slices subsequently obtained by them at the end of the game.

b. In this instance, there is a probability of the risk that if the total sum of the number slices chosen by both players exceeds 8 then the player choosing a relatively greater number of slices would not get anything in case both players choose different number of slices. Now, if the sum total of the number of slices selected by both players is greater than 8 and both players choose the same number of slices, then both would get 4 slices each. Therefore, in this case, each player would have a rational tendency to choose 4 pizza slices as note that, even if the other player chooses more any number of slices more than 4 slices, the sum total of the number of slices selected by both players would exceed 8 and the player choosing 4 slices would automatically get his or her desired number of slices and the other player would be eliminated from the game. On the other hand, even if the other player or the opponent player also chooses 4 pizza slices then both players would get 4 slices anyways which is guaranteed and if the opponent or other player selects any number less than 4 pizza slices then the sum total of the pizza slices chosen by both players automatically becomes less than 8, in which case, the player choosing 4 pizza slices would unquestionably get 4 slices as desired by him or her. Therefore, considering all the three possible scenarios presented in the question, choosing 4 pizza slices would be the most optimal or rational strategy for both players in this instance. Hence, a pure strategy Nash equilibrium of this game could be each player choosing 4 slices of pizza each and getting 4 slices each subsequently.