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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766 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
59 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 ...
5
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1answer
248 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
192 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 ...
3
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1answer
201 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 ...
4
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0answers
142 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 ...
7
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2answers
209 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. ...
4
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1answer
630 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
94 views

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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0answers
63 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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0answers
31 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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0answers
82 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 ...
3
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0answers
65 views

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 ...
3
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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 ...
3
votes
2answers
116 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
142 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 ...
12
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3answers
379 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. ...
5
votes
1answer
90 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 ...
7
votes
2answers
346 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. ...
4
votes
1answer
105 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 ...
2
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2answers
113 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
89 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 ...
2
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5answers
326 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
202 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 ...
0
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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 ...
6
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1answer
145 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
128 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 ...
2
votes
1answer
234 views

Is there an example of zero-sum game that has a Nash equilibrium which is not subgame perfect?

As a refinement of Nash equilibrium, it is known that not all Nash equilibria are subgame perfect. But it seems to me in zero-sum games of perfect information, Nash equilibrium coincides with subgame ...
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1answer
109 views

Two Player Game Useless Strategy

Let's consider the variant of dominated strategy which is the pure strategy that is not a best response to any mixed strategy of the opponent (two player game). Intuitively it sounds like more ...
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0answers
119 views

Algorithm to verify that a weak Nash equilibrium is an ESS, or a strict Nash equilibrium

Is there any algorithm that might assist me in checking whether a weak Nash equilibrium in a signalling game is also an Evolutionarily Stable Strategy, or a strict Nash?
4
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0answers
85 views

Linear Independence Game

Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
1
vote
1answer
59 views

Concept of efficiency in auctions

I have some confusions about the concept of "efficiency" in auction theory. One interpretation is that an auction is efficient if it maximizes the social-welfare. But social-welfare is not well ...
1
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1answer
70 views

Question on the use of induction in the Electronic Mail Game

In Rubinstein's Electronic Mail Game, Player I and Player II's strategies take the form as $s_i : \mathbb{N} \to \{A,B\}$, $(i =1,2)$. Rubinstein shows that the pair of constant functions, $s_1(t_1) ...
2
votes
1answer
90 views

Is there a game of perfect information that only has a mixed strategy equilibrium?

It is known that certain simultaneous games, say, matching pennies with no pure strategy equilibrium, have a mixed strategy equilibrium. Is simulaneous move at some stage a necessary ingredient of ...
2
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0answers
308 views

Is quantum game theory reducible to classical game theory? [closed]

Modnote: This question was manually migrated (closed and crossposted) to MathOverflow by request of the OP. Quantum game theory is an extension of classical game theory to the quantum domain. It ...
1
vote
1answer
232 views

Maxmin strategy and Nash equilibrium

It's well known fact that maxmin strategy in Nash equilibrium in the two-players zero-sum finite game, but to prove it? How to show that maxmin strategy is actually Nash equilibrium in the case of ...
3
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1answer
55 views

circular table game

"Suppose there are $n$ chairs around a circular table that are labelled from $0$ to $n-1$ in order. So chair $i$ is in between chairs $i-1$ and $i+1$ mod $n$. There are two infinitely smart players ...
1
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1answer
51 views

Question on the construction of mapping from space of strategy profile into itself in Nash(1951)

To appeal to Brouwer fixed point theorem, Nash(1951) constructed a continuous mapping $\operatorname{T}$ from strategy profile space into inself: For player $i$, the probability of a pure strategy ...
6
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2answers
103 views

Does there exist a finite fair gamble game with one dishonest coin?

I am thinking, maybe a well known problem, of whether there exists a fair gamble game for two persons by tossing one dishonest coin that will always stop(one winner is selected) at no more than $N$ ...
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0answers
161 views

Stable Matching Optimal Strategy Existence

There is a famous stable marriage problem. It's well known that the standard algorithm for stable marriage problem proposed by Shapley and Gale is man-optimal, men get a best according to their ...
0
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1answer
84 views

How to define a mixed strategy in a game with a countable action space

Suppose in a two player zero-sum game, player I chooses a number $m$ from $\mathbb{Z}$, and player II chooses $n$ from $\mathbb{Z}$. If $m-n =1$, then player I recieves a payoff of $1$, while ...
4
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0answers
125 views

Unexpected hanging paradox maxmin strategies

I have a question about strategies of the players of Unexpected hanging paradox (I am very sorry for a long topic, topic exist already for a while, during this time I try to develop idea how to solve ...
0
votes
1answer
73 views

Finding a dual Linear-Program

We are trying to prove Von-Neumann's MINIMAX Theorem namely $$\max_{x\in\Delta_{n}}\min_{y\in\Delta_{m}}y^{T}Ax=\max_{x\in\Delta_{n}}\min_{1\leqslant i\leqslant n}(Ax)_{i}$$ (Here $\Delta_k$ is the ...
2
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2answers
676 views

Minimax solution for Zero-Sum Game

I try to understand the way to finding the minimax solution to zero-sum game. The following example is takes from Wikipedia. Minimax Wikipedia: The following example of a zero-sum game, where A ...
0
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1answer
159 views

gain-floor and loss-ceiling in minimax theorem

I am little bit confused by the lemma preceding to minimax theorem in game theory . I use the following material for studying GAME THEORY AND THE MINIMAX THEOREM. gain-floor definition. A ...
6
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1answer
124 views

Find the value of a function with definite integrals

I am trying to understand a paper of Maynard Smith (1974), that connects biology with game theory. I don't want to overwhelm you with useless stuff, but I have this definite integrals: ...
4
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2answers
117 views

Does Arrow's Theorem apply when choosing a single best candidate?

According to Wiki, Arrow's Impossibility Theorem proves that we cannot create a social welfare function that obeys unanimity, non-dictatorship, and IIA. However, in real elections, we want to choose ...
3
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1answer
521 views

Converting a game in extensive form to normal form

I have some difficulties in representing the following game in the standard form. Game: two players game is represented as a game tree (in extensive form), a game tree is a full binary tree, both ...
12
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0answers
389 views

“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
3
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1answer
37 views

Is it necessary to state that $y_i \leq 1$

In a class test for Linear Programming, my professor deducted some marks because I missed the condition $y_i \leq 1$ in the mixed strategy games solution. $ y_i $ stands for the probability of any ...