Challenge 1: A Beautiful Mind
1.)
Let’s assume that one player, for example Sheldon, has a probability p>0 to guess 100. In this game all of the other players have the same probability p>0 to choose 100 as well. Our player in question (Sheldon) thus can improve his winning chances by guessing any integer below 100. This increases his winning chances if all the other players stick to their guess of 100. Since every player in the game has the same knowledge of these game properties, they all try to outguess each other by guessing a number below 100. When we look at the next smaller number, 99, the appropriate reaction would be to guess 98. This process of eliminating the weak strategies continues until the lowest number is reached, which in this case is 1.
Hence the Nash equilibrium for this game is when each player guesses 1. In this case, they can be certain that they chose the best strategy according to any other player's guess. This cycle which we describe reflects the theory that “at a Nash equilibrium outcome, each player is doing the best it can, given the strategies of the other players” (Besanko, Dranove, Shanley, & Schaefer, 2013).
Empirically this is proven by the problem called the “p-Beauty Contest”, which was developed by John Maynard Keynes (Tadelis, 2013).[pic 1][pic 2]
As we can see from the formula the set of winners W ⊂ N is defined as the players who guessed the closest to a certain amount of the average. If we apply the formula to Sheldon’s situation we arrive at the following formula, where the winners are defined by the players who guess closest to of the average.[pic 4][pic 3]
[pic 5]
The game is called the “p-Beauty Contest” because the players are not trying to guess in accordance with everyone else, instead, one is trying to guess p-times the average of the sum of the guesses.
2.)
(1)
The above mentioned state is not a Nash equilibrium. Leonard´s last bid was $1.99. If Penny wouldn´t bid more, she would lose her bid ($1.98) to Sheldon and thus raise her bid to $2. In this case, she would now make zero profit (win $2 but pay her bid of $2 to Sheldon). However, Leonard now has an incentive to bid $2.01, to lose only $0.01 ($2 submission fee & bonus minus his bid of $2.01) instead of losing his prior bid worth $1.99. The game continues like this indefinitely because the difference between any number X > 2 and 2 (winning the bonus but losing the bid) is lower than the absolute value of X (losing the bid). In other words, there will always be an incentive to raise the bid because in any situation above $2 the “winner” is exactly $1.99 better off than the “loser”. After surpassing the $2-bid, Leonard and Penny fight to minimize their losses.