Tagged Questions

The study of competitive and non-competitive games, equilibrium concepts such as Nash equilibrium, and related subjects. Combinatorial games such as Nim are under [tag:combinatorial-game-theory], and algorithmic aspects (e.g. auctions) are under [tag:algorithmic-game-theory].

73 views

Game theory - Finding Nash Equilibria

i have to solve these questions. I already came up with a solution. Can anyone look over it and state their opinion? Many thanks in advance :) CHALLENGE I 1. Consider the following game between ...
50 views

How to prove that the zero sum game's optimal (security) strategy do not change when payoff matrix increase by a constant factor

I am not sure how to construct a proof for something so obvious, can someone give me some pointers on proving the following Suppose a constant K is added to each element of a pay-off matrix ...
43 views

Best Strategy in a Non-Cooperative Game of Perfect Information (Pure Strategies only)

Suppose there are two players in this game, and each player has $4$ dollars prior to making his move. Each player has as his strategy space the ability to submit an integer number of these $4$ initial ...
26 views

Game Theory rationalizable strategies that are all the same

Suppose player 1 has strategies u, m, and d, and player 2 has strategies A, B, and C. The strategies A, B and C are all identical (payoff equivalent). Can any of them be eliminated using the ...
46 views

Playing rock paper scissors over online chat.

Is there a way to play rock, paper and scissors fairly over internet chat? By this, I mean that both players cannot play their hands simultaneously, one of them has to go first and the second player ...
39 views

What is the optimal losing move?

I had a hard time trying to find the best-suited stackexchange site to ask this question. I'm still not sure whether this is the right place, so please guide me to the right one if you think this is ...
42 views

Optimal strategy to escape spotlight

Here is the setup. A prisoner is being held in the center a circular yard with radius $r$ and can run in any direction at some velocity $v$, there is a spotlight which illuminates a line on the circle ...
25 views

One Million Proficiency Efficiency Calculations

I have been playing the "Million" game on my apple device after discovering it through the trending applications in the search function. After beating the game I began to wonder to myself if there was ...
26 views

Find the mixed strategy Nash equilibria in the investment race

This is Exercise 35.2 in Rubinstein's "A Course in Game Theory". This problem is very difficult. The lecturer gave us the answer but it's very hard to understand. I paste the problem here: Two ...
95 views

Coin flipping game - probability and game theory

I'm exercising for a test and I've been trying to solve this problem. John bets 1 or 2 dollars before flipping a coin. If the result is heads, he wins the sum he has bet. If the result is tails, he ...
118 views

Is there any winning strategy? 2015 and Game with marbles!!!

Two players, Alex and Brad, take turns removing marbles from a jar which initially contains 2015 marbles. Assume that on each turn the number of marbles withdrawn is a power of two. If Alex has the ...
27 views

Making sense of an Expected Utility of a Mixed-strategy Profile definition

I came across a definition of Expected Utility of a Mixed-strategy Profile in Brown's and Shoham's "Essentials of Game Theory: A Concise, Multidisciplinary Introduction" where: "Given a normal-form ...
220 views

Maximize the Earning

There are two thieves with a total of 1 billion ($10^9$) dollars in a bank vault. Now they must decide how to divide the booty. But there is one problem: the thieves have only $M$ minutes to leave the ...
179 views

How to divide the money based on the below conditions? [closed]

Two cheeky thieves have collectively decided to rob a bank. They have carefully counted a total of exactly $100$ dollars in the bank vault. Now they must decide how to divide the booty. But there is ...
39 views

Shapley values as a hash/compression of a game

Computing a Shapley value, we are mapping the set of coalition games on $N$ to a vector of $N$ elements: $$\phi: \; \mathbb{R^{2^N}} \to \mathbb{R}^N$$ In a sense, this is compression, or more ...
68 views

Multiplying Single Digit Numbers to Get Product >1000

This is yet another Alice and Bob problem. Alice and Bob are playing a game on a blackboard. Alice starts by writing the number $1$. Then, in alternating turns (starting with Bob), each player ...
68 views

Good books to learn Combinatorial Game Theory? [duplicate]

I am currently doing my IB Diploma and we are supposed to make an extended essay on a subject of our choice- and i chose math. my research question is- "how to derive a perfect strategy to always win ...
37 views

Game theory: solutions concepts if each player has different utility

Assuming, I have a network game (where the players are e.g. users in a network), how is it possible to model user-specific a-priori information (which results in different utility functions for each ...
25 views

Question about usage of $\leq$ in definition of Nash equilibrium

Quick definition: Given $g$, a strategy N-tuple $u^* = (u_1^*,...,u^*_N)$ is said to be a Nash equilibrium if: $$J_i(u_i^*, u^*_{-i}) \leq J_i(u_i, u^*_{-i}), i \in N$$ where $J$ is ...
49 views

What is the quickest way to find Nash equilibria in two player bimatrix game?

Suppose the cost/penalty matrix of a game is given as: $$M = \begin{bmatrix} (-5,-5) & (0,0) \\ (0,0) & (-3,-3) \end{bmatrix}$$ Then the game as two equilibria $(u_{11},u_{21})$ and ...
48 views

In what ways would a course on convex optimization be useful in game theory?

From talking to several other people in the past, and referencing Quora, it seems that convex optimization is really a tiny subset of game theory in that it only models the behavior of one single ...
30 views

my text defines player set as: In N-player game $g$, each non-terminating node is partitioned into $N+1$ sets $g^0, ... g^N$. These are player sets. However it makes no attempt to identify ...
29 views

Devising a method to arrive at a square only being able to move right and up

I was recently presented with a problem where I have a 10x10 grid and a marker that starts at position (1, 1). Two players are able to manipulate the marker by moving it as far right, or (exclusive) ...
68 views

Reducing TIC-TAC TOE State Space by using Symmetry in Artificial Intelligence

Im learning Heuristics in AI.I see that for brute force search there are 9! states.But the textbook says that first 3 levels are reduced by symmetry.How does that work?
20 views

A question in Osborne-Rubinstein A course in game theory,

I am having trouble with Rubinstein's electronic email game (proposition 83.1) in the textbook(first edition). My question: Line 7 starting from the "proof", "...player 2's expected payoff is at ...
130 views

How many legal states of chess exists?

I have a fairly simple question. How many legal states of chess exists? "Legal" as in allowed by the rules and "state" as an unique configuration of the pieces. I'm not asking for the number of ...
19 views

numerical solution for nash equillibrium

I have the following setup. $\pi_1=f_1(q,r)$ and $\pi_2=f_2(q,r)$ are the real valued payoff/profit functions of the two players. Player 1 gets to pick $q$ and player 2 gets to pick $r$. I also know ...
43 views

Probabilities in this blackjack variation

Let's say I play blackjack (52 cards, figures count for 10, aces count for 1 or 11) and alone (no dealer). The cards I use for one particular game are always removed at the end of that game and won't ...
91 views

“Composition” property in cooperative game theory?

I am trying to find a property which can help to analyse the composition of a cost/profit division and which allocation rules (e.g., Shapley value or nucleolus) would satisfy it. In short, the idea ...
76 views

Tzaloa 2015 game problem (piles with $1,2,4 \dots 2^{19}$ coins each)

We have $20$ piles with $1,2,4,8\dots 2^{19}$ coins repectively and two players. In each turn a player must select five piles that have at least one coin and remove exactly one coin from each. Player ...
67 views

Why does the 1st player in this subset take-away game always have a winning strategy?

This is a HW problem of mine that I cannot, for the life of me, figure out. There is a take-away game where there are a number of elements A, and the person that wins is that last person to remove a ...
27 views

Nash equilibrium for two players game.

Consider a game for two players, say "Player A" and "Player B". The two sets of strategies are denoted by $A$ and $B$, available to the players. Consider a symmetric situation where the players have ...
329 views

Expected travel of random walk in arbitrary game with multiple payouts

As explained here, the average distance or 'travel' of a random walk with $N$ coin tosses approaches: $$\sqrt{\dfrac{2N}{\pi}}$$ What a beautiful result - who would've thought Pi was involved! ...
33 views

When solving a system of equations for a game theory question, can the solutions be negative?

I have a homework question on solving a game matrix geometrically. $m =$ $\begin{bmatrix}1 & 11\\7 & 2\end{bmatrix}$ (after adding the constant $k$ to ensure it's a positive matrix) The ...
59 views

Find a Nash equilibrium solver

The solvers I know so far are designed only to allow payoffs as given numbers. But is there a solver allowing users to type payoffs as variables?
81 views

Finding Nash equilibrium in game with random event at event tree?

I have posted a question about finding the NE of sequential game with imperfect information. It is lucky that the game can solve could be dealt with could be dealt with by a simpler argument. Here is ...