Articles of game theory

Big Balloon Game

The problem In this game, you are given empty balloons one by one, and for each balloon you are to inflate it with air until you are satisfied. If it does not burst, you gain happiness points proportional to the volume of air in the balloon (say 1 point per ml). If it bursts, you […]

Understanding common knowledge in logic and game theory

For $k = 2$, it is merely “first-order” knowledge. Each blue-eyed person knows that there is someone with blue eyes, but each blue eyed person does ”not” know that the other blue-eyed person has this same knowledge. (from http://en.wikipedia.org/wiki/Common_knowledge_(logic)) I am not getting this. If there are more than one people that have blue eyes, […]

What is the sprague-grundy value of these games?

This is a follow-up question of my previous question : Optimal strategy for this Nim generalisation? Consider the following game: There are a number of piles of stones. On each turn a player can remove as many stones he likes (at least 1) from up to N piles (at least 1). It is allowed to […]

Nash Equilibrium for the prisoners dilemma when using mixed strategies

Consider the following game matrix $$ \begin{array}{l|c|c} & \textbf{S} & \textbf{G} \\ \hline \textbf{S} & (-2,-2) & (-6, -1) \\ \hline \textbf{G} & (-1,-6) & (-4, -4) \end{array} $$ which corresponds to the well-known prisonder’s dilemma. Now a Nash Equilibrium by using pure strategies would be (G,G) cause by choosing them neither can improve his […]

What is the (expected) outcome of this hybrid auction?

A certain hybrid auction can be accurately modelled as follows. There are $n$ risk-neutral, rational participants $i=1,2,\ldots,n$, and a guy called Zerro: $i=0$. Each, except Zerro, has a private value of the good (only known to themselves) $v_i\sim_\mathrm{i.i.d.} \mathrm{UNIF}(0,1)$. An English auction (EA) is held. Zerro is allowed to, and does, open the bid at […]

Weighted War – Game of Mind and Probability

Game Weighted War is a game of bidding, where: Both players have cards valued from $1$ to $11$ in their hands There is a third pile of cards from $1$ to $11$ face down on the table and shuffled, with one random card being removed from it at the beginning of the game Each beginning […]

Very fascinating probability game about maximising greed?

Two people play a mathematical game. Each person chooses a number between 1 and 100 inclusive, with both numbers revealed at the same time. The person who has a smaller number will keep their number value while the person who has a larger number will halve their number value. Disregard any draws. For example, if […]

Why can a Nim sum be written as powers of 2?

I have this confusion. Why do we express a nim sum as powers of 2 and why do nim sums cancel in pairs of 2 only? For instance, let’s take the nim game(6,10,15) Now clearly *6 = * $2^2$ + * $2^1$ *10 = * $2^3$ + * $2^1$ *15 = * $2^3$ + * […]

Why the set of outcomes generated by a fixed strategy of one player in Gale-Stewart game is a perfect set?

In the proof that there is a payoff set $X$ such that the Gale-Stewart game is not determined(see here, Proposition 3.1.). I don’t know why $X$, the set of all outcomes generated by a fixed strategy of one player, constitutes a perfect set. I can see it’s true, if the fixed strategy is a constant […]

Pirate Game (modified)

http://en.wikipedia.org/wiki/Pirate_game What happens if you remove the order of seniority? Whenever a pirate dies, you randomly pick the next pirate to propose a distribution. Here’s my solution for 5 pirates: A: 48 B: 26 C: 26 D: 0 E: 0 How about for $ n $ pirates and $ p $ gold coins?