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

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Convex combination of correlated equilibria

Prove that any convex combination of correlated equilibriua is also a correlated equilibrium.
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Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
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1answer
77 views

A combinatorial game about stones

There are some piles of stones. Two players move in turn. One can remove a stone from a pile or merge two piles in a move. The player that removes the last stone wins. With the number of stones in ...
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0answers
149 views

Mathematical reason for 2-player turn-based games

I've been reading Games, Puzzles, and Computation which analyzes games through game theory and complexity theory. The authors introduce something called "Constraint Logic" as a way of modeling games ...
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3answers
3k views

A non-losing strategy for tic-tac-toe $\times$ tic-tac-toe

Consider a 9 by 9 matrix that consists of 9 block matrices of 3 by 3. Let each 3 by 3 block be a game of tic-tac-toe. For each game, label the 9 cells of the game from 1-9 with order from left to ...
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108 views

Optimal auction for risk averse seller

Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim ...
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1answer
312 views

Air Strike Game

This is an Air Strike Game with the solution, I have added some questions regarding the solution and I would appreciate if someone could answer them. Army $A$ has a single plane with which it can ...
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1answer
225 views

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 ...
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1answer
110 views

Expected revenue in first bid auction.

We find expected revenue in first bid auction by following method. let us say $V_1$ and $V_2$ denotes maximum amount that player 1 and player 2 willing to pay. $V_1,V_2 \in [0,1]$ In case when we ...
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1answer
75 views

Generlized Büchi Games and Closed under superset Muller Games

For a unique infinite play $p$ in a 2-Player game $G=(V_0,V_1,E)$. Let $$ \inf(p) \subseteq V_0 \cup V_1 $$ be the set of vertices which occur infinitly often in $p$. Generlized Büchi (GB) Games ...
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301 views

Exercise in Mechanism Design

I found an exercise with solution in the field of Mechanism Design. The problem is I don't understand the solution. Exercise. Use the characterization of incentive compatible direct-revelation ...
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2answers
146 views

Fastest way to get \$1 million of two different currencies in a video game

This question actually relates to a video game, I came across the scenario and I realized I had no idea how to go about solving something like this or even what branch of mathematics it falls under. ...
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3answers
131 views

Is there any research field dedicated to estimating a “game” itself in game theory?

Game theory stuffs usually provide how a "game" works and then tries to figure out solutions - but I am wondering if there is any research field dedicated to estimating the full rules of a game. So ...
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2answers
338 views

Dominant Strategy in Table Games

I have some basic background in game theory, but still there are exist simple questions that I cannot answer for sure. Whether Tic-Tac-Toe game has a dominant strategy? May be only one of the ...
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0answers
61 views

Any known strategies for toads and frogs?

Are there any heuristic strategic for playing Toads and Frogs known? I reckon the optimal playthrough may be hard to achieve due to the game being NP-hard but at least something that regularly ...
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1answer
271 views

Mixed Strategy Simplification

I am very consufed by the following problem. Show that for any fixed $\epsilon > 0$ and any 2 player game with all non negative payoffs, there is an $\epsilon$-approximate Nash equilibrium such ...
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3answers
736 views

Game Theory: Is there an “unexploitable” strategy in No Limit Holdem?

Do "unexploitable" strategy exist in No Limit Holdem? By this I mean frequency-based mixed strategy that has non-negative expected payoff against any other strategy (let's assume the game is ...
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3answers
752 views

Three against the devil: a combinatorial game

A team of three sinners plays a game against the devil. They confer on strategy beforehand; then they go into three separate rooms, and there is no more communication between them. The play in each ...
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1answer
232 views

Is convergence to a Nash Equilibrium dependent on turn order?

Is convergence to a Nash Equilibrium dependent on turn order? Namely, if you change the turn order or switch between synchronous (all players move at once) and asynchronous turns can the outcome ...
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1answer
186 views

Approximate Nash Equilibrium

I am sort of confused by the notion of approximate Nash equilibrium. I will try to express my confusion in the following exercise. Problem. Is it true that for every two player game where every ...
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4answers
2k views

rock, paper, scissors, well

Everyone knows rock, paper, scissors. Now a long time ago, when I was a child, someone claimed to me that there was not only those three, but also as fourth option the well. The well wins against rock ...
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1answer
128 views

Is there no Pareto-optimal Shotgun game?

I thought about an example game to use to illustrate what "Pareto-optimal" means, and I can't think of an outcome of Shotgun game (rock, paper, scissors) played by three players that would be ...
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2answers
283 views

Does introducing penalties for getting true/false questions incorrect result in higher skill penetration (less luck/variance)?

A student is asked to answer 50 true/false questions and he would get 35 right and 15 incorrect if he had to put his best guesses for each question down. Now, for each question he has a certain ...
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3answers
850 views

prerequisites for understanding game theory

I am from programming background but with very limited knowledge of maths. I am very much eager to learn and apply game theory to understand dynamics of International Politics and economics. But I am ...
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1answer
66 views

Calculating the core of a game

In a coalitional game n miners find equal blocks of gold. Two can carry one piece home. The payoff of a coalition S is $\nu(S)=|S|/2$ if $|S|$ even and $(|S|-1)/2$ for $|S|$ odd. Determine the core ...
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1answer
256 views

When do $\epsilon$-Nash equilibrium strategies converge to Nash equilibrium strategies?

Suppose I have a game on $n$ players and a sequence of strategy profiles $(s_1^{(1)},\dots,s_n^{(1)}), (s_1^{(2)},\dots,s_n^{(2)}), (s_1^{(3)},\dots,s_n^{(3)}), \dots$. Each ...
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1answer
198 views

Is game theory a part of math? [closed]

I'm going to write project paper for my course, "The History of Math". My field is 'game theory'. However, I'm in doubt that game theory is really part of math, since it comes from economics (for ...
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1answer
218 views

Existence of asymmetric equilibria in the dollar auction game

Consider a game in which an auctioneer sells one dollar to the highest bidder. The high bidder wins the dollar, but every bidder pays their bid. Concretely, assume that there are two bidders ...
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143 views

algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
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2answers
217 views

What is the name of a game that cannot be won until it is over?

Consider the following game: The game is to keep a friend's secret. If you ever tell the secret, you lose. As long as you don't you are winning. Clearly, it's a game that takes a lifetime to win. ...
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1answer
772 views

The Notion Of Degenerate Two Player Game

I try to get the intuitive understanding of the notion "degenerate two player game". Definition. A two-player game is called non degenerate if no mixed strategy of support size $k$ has more than $k$ ...
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0answers
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Applications of Scoring Play Combinatorial Game Theory

I'm currently looking into economic applications of scoring play combinatorial game theory. Details of the theory can be found in this paper. http://arxiv.org/abs/1202.4653 A friend of mine ...
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66 views

Game tie probability

If we have 3 players, playing extreme RPS game like image below: How much tie probabilities? Thanks
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34 views

Open infinite games and payoff functions

Let $A,B\subseteq\mathbb{N}$ and $d(A,B)=\sum_{n\in A\Delta B}2^{-n}$, where $A\Delta B = (A\cup B)\setminus (A\cap B)$, be a metric on subsets of the natural numbers. I'm asked to show that for ...
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84 views

Infinite games and pay-off functions

I'm asked to prove that, in a version of the Gale-Stewart game where the two players alternately pick zero or one, the pay-off function cannot be continuous. From what i have read, the pay-off ...
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0answers
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Terminology questions about a game where one may “save his progress” at the cost of a turn.

The game is for $p$ players who each start at square $1$. Each turn, a player can either roll an $m$-sided dice or place a marker on his current square. If he rolls $x\in\{2,\ldots, m\}$, he ...
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1answer
81 views

Stability under supremum of sets of social choice function with single peaked preferences

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ...
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2answers
117 views

probable squares in a square cake

There is a probability density function defined on the square [0,1]x[0,1]. The pdf is finite, i.e., the cumulative density is positive only for pieces with positive area. Now Alice and Bob play a ...
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1answer
143 views

You are Johnny Depp 3!

An extension of this question. As @Jared stated in his answer the solution is: we assume that the head pirate chooses between multiple possible proposals that maximize his profit by rewarding ...
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3answers
387 views

A number guessing game

Alice chose a positive integer $n$ and Bob tries to guess it. In every turn, Bob will guess an integer $x$ $(x>0)$: If $x$ equals $n$, then Alice tells Bob that he found it, and the game ends. ...
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1answer
94 views

AI strategies for losing positions [closed]

I have a card game that I am analyzing with Maple. Actually, it's a series of card games, one for every parameter k, where k is a natural number (representing the number of ranks of cards used in the ...
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2answers
349 views

You are Johnny Depp 2!

An extension of this question repeated below. A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. ...
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1answer
111 views

Finding maximum score in a “bubble pop” game

Consider the following game: there is a n×n field, where each cell is randomly coloured in one of m colours. Let a group of cells be a set of same-coloured cells s.t. every cell in a group has at ...
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2answers
115 views

nash equilibirum help! seems tricky

Any advice for finding all nash equilibrium for this symmetric game? (B,B) looks like one but I feel like there are more. I tried looking for strictly dominant strategies, but only A weakly dominates ...
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1answer
91 views

Optimal strategy for a mixed game

I'm trying to understand what is the optimal strategy for a mixed game. I can illustrate the game as a trading system where you can go Long or Short. Going Long will give 80% win rate and going short ...
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5answers
336 views

Game Theory Question about Financial Markets

This is a recent quote from one of the outstanding bond portfolio managers: "First of all, for every buyer there is a seller. Therefore, in order for someone to sell their bonds and buy stocks means ...
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0answers
211 views

Strictly Dominated and Never Best Response in LP

There is a well known notion of Strategic Dominance in Game Theory. I am interested in elimination of strictly dominated strategies by Linear Programming and in LP for definition of never best ...
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1answer
107 views

Using limit argument with non-continuous social-choice functions

This question is related to another question of mine Invariance of strategy-proof social choice function when peaks are made close from solution, and it revolves around the use of limit arguments with ...
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1answer
149 views

Invariance of strategy-proof social choice function when peaks are made close from solution

A question emerging from reading Schummer, J., & Vohra, R. V. (2002). Strategy-proof Location on a Network. Journal of Economic Theory, 104(2), 405–428. The setting is as follows: A finite set ...
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2answers
132 views

Game Theory: Penalty Shot Game

Given a game matrix for the penalty shot game: (1/2,-1/2) (-1,1) (-1,1) (1/3,-1/3) What is the minimax ...