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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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 ...
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2answers
97 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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89 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 ...
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1answer
67 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 ...
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98 views

A Gale-Stewart version of hawk-dove game

Suppose, in a two-player $\omega$-game of perfect information, player I or player II's action space at an arbitary stage $i$, $A_i$, is fixed as $\{H, D\}$ ($H$ and $D$ denote Hawk and Dove ...
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85 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 ...
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1answer
61 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 ...
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2answers
368 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 ...
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1answer
80 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 ...
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1answer
106 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: ...
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2answers
99 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 ...
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285 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 ...
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203 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 ...
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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 ...
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2answers
88 views

$n$-player version of Zermelo's Theorem

Zermelo's Theorem states that "Every finite zero-sum 2-player game is determined (one of the two players has a winning strategy)." I was wondering if anyone has investigated the generalization of this ...
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3answers
121 views

How to write “the parameter maximizing the maximum of the maximum value of two functions continuous in the domain of maximization”

Say you have $f(x),g(x)$ continuous where they need to be and you want to express the following: Give me the biggest value of $f$ for $x \leq X_f$ , give me the biggest value of $g$ for $x \leq X_g$, ...
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1answer
70 views

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 ...
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244 views

Finding the payoff matrix of a game

A two player zero-sum game can be represented by a $m\times n$ payoff matrix $M$ having $m$ rows and $n$ columns with values in $[0,1]$. The value $M(x,y)$ represent the payoff given to player $1$ ...
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2answers
144 views

Find a stronger AI for a game, or even a winning strategy

It is a two player game. First we have a eight by eight matrix with a white stone on the (1,1) entry and a black stone on the (8,8) entry. Then alternatively Player 1 and Player 2 move the white stone ...
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2answers
158 views

Is risk-taking a zero-sum game?

I learnt that trading is a zero-sum game since profits take out losses while holding your assets may not always be a zero-sum game when time passes and everybody's assets theoretically can grow in ...
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223 views

$n$ players of paper scissor rock

Suppose there are $n$ players $(3\leq{n})$ showing Paper, Scissor or Rock simultaneously. If there is no winner then there is no payoff to any player. If there are winners and losers (e.g. $k$ ...
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97 views

How to calculate Team Strength for future prediction?

You are given with $4$ players name, namely Player $A$, Player $B$, Player $C$ and Player $D$. These players are grouped into two teams with two players each. A Game is played between the two team.For ...
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1answer
115 views

Games for which the Lemke-Howson algorithm provides incomplete results

I am exploring a large number of 2-player games. The Lemke-Howson algorithm is computationally very reasonable, and is able to find many equilibria. On the other hand, I know that there are equilibria ...
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85 views

A negotiation problem

We have a list of players $P_1\ldots P_n$ and a set of $n$-vectors $S$, which always contains the $0$-vector (representing no deal). Each vector $(s_1,s_2,\ldots,s_n)$ in the set represents a deal ...
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117 views

Find profit maximizing profit and quantity given willingness to supply curves and a merger

This is a homework problem, but I'm at my wit's end. I don't even know where to start on this, but I've tried a number of strategies. Consider a regional market for wholesale electricity where ...
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1answer
84 views

ODE with global inequality constraint on derivative

The problem is solving the ODE $f''(x)+a_1f'(x)+a_2f(x)=g(x)$ with boundary conditions $f(c)=h(c)$, $f(d)=h(d)$, where $c,d$ must be such that $f'(x)\geq b\;\forall x\in[c,d]\subseteq\mathbb{R}$. The ...
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2answers
1k views

Cournot-Nash Equilibrium in Duopoly

This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way of solving the following question: There are 2 firms in an industry, ...
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0answers
80 views

Understanding Blackwell's Approachability Theorem

I'm not super solid on my linear algebra, so I am getting lost in the discussions of halfspaces. Can someone give me an intuitive explanation (possibly with a concrete toy problem) of Blackwell's ...
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4answers
120 views

How to win a game like this?

My teacher started a game like this "everyone hands in a number in [0,99] in the next class, and the winner is the one with the number that is closest to half of the average of all submitted numbers". ...
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1answer
73 views

Criterium for Nash-Equilibrium from Nash's Paper

I am reading Nash's original paper "Non-cooperative games" from 1951, which could be found here: Non-Cooperative Games, Nash (1951) Now I have a question to criterion (2) on the second page. There ...
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2answers
1k views

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) ...
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50 views

Second Price Auctions - Duplicate Bids and Attracting Bidders

In a second price auction, what occurs if two bidders bid the exact same amount on a non-divisible item. For instance, let's say these are the bids for object A: ...
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58 views

more examples on games

As in the example of the game of $Hex$, it needs some tools from algebraic topology to prove that there is exactly one winner in the game I wonder if there are more examples on games to be ...
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2answers
433 views

GameTheory, Solve for optimal strategies by solving a system of linear equations

In a book on game theory I saw the following example of a game, a modified version of Roshambo (or Rock-paper-scissors), which is described by the following payoff-matrix: $$ \begin{array}{c|c|c} ...
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1answer
113 views

What is the general formula for electoral districts tying.

I apologize if this question is a bit of a read. (You might want to get a frosty beverage.) Professor Alan Natapoff of MIT demonstrated, if 9 Voters are districted into 3 electoral districts of 3 ...
6
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3answers
327 views

Game theory games with very counter-intuitive results?

I have heard of an interesting game that produces a very counter-intuitive result. It is an auction of a 100 dollar bill, but one in which both the first person in the auction and the second need to ...
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1answer
47 views

Theorem that stable equilibria in iterated games are equivalent to coalition-based static equilibria

Consider an $n$-player nonzero sum finite game $G$. I have a vague recollection of a wonderful paper proving an equivalence between (1) steady state Nash equilibria of $G$ played countably many times ...
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1answer
378 views

Nash equilibria and best response functions

a) Let $G=(A,u)$ be a strategic game such that, for each $i \in N$ $A_i$ is a nonempty, convex, compact subset of $R^{m_i}$ $u_i$ is continuous For each $a_{-i}$, $u_i(a_{-i}, . )$ is quasi-concave ...
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2answers
211 views

Nim addition- binary addition without carrying

A nim addition table is essentially created by putting, in any cell, the smallest number not to the left of the cell and not above that cell in its column. However, I know for a fact that nim addition ...
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1answer
177 views

Prove set of Nash equilibria is closed?

Is this even possible with just the formal definition of a Nash equilibrium, that is, without any additional conditions, such as the utility function is continuous? Thanks.
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1answer
365 views

Gibbard–Satterthwaite Theorem versus Arrow Theorem

Arrow Theorem is a very classical result in social choice theory, stating very roughly that any reasonable voting procedure is either dictatorial or subject to tactical voting. More precisely, there ...
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1answer
249 views

What is the optimal strategy for this game?

You are playing a game where you put in a certain amount of money $m$. A random number in $[0, 1]$ is chosen. If the number is greater than $p$, you now have k% more money, otherwise, you lose all ...
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1answer
108 views

Modellering a Integer Linear Program

Warning; !! Long post !! Note; This is not a homework assignment, but rather an old exam question I'm trying to figure out. If you read on, you'll notice that I've put quite some work in on it ...
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1answer
144 views

Goofy problem: Optimal bet with nearly no knowledge

A year or so back, on the verge of falling asleep, I thought up this question: You have come to me ready to gamble. I have two envelopes on the table, one containing the amount of my bet, and one ...
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173 views

Optimal Mixed Strategy for matrixes of size NxN

Say I have a Game Matrix of size $(N+1) \times (N+1)$, alike this one below \begin{array}{cccccc} 0 & 1 & 2 & \ldots & N-2 & N-1 & N \\ 1 & 0 & 1 & \ldots & ...
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1answer
97 views

Deducing probability of an event, when opponent's type is uncertain

Suppose, two players I and II, given a state space of three states$\{a,b,c\}$ with a common prior, $p(a) = p(b) =p(c) =1/3$, are endowed with two partitions of state space, $\mathscr{P}_\text{I} = ...
2
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1answer
108 views

convergence of functions on probability measure

I am studying a problem in game theory, but I am lacking on knowledge to deal with a continuum of distribution functions convergence. $\mathfrak{F}([0,1])$ is the set of distribution functions over ...
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81 views

Game theory question- information quality maximisation, opinions of the question

I am developing a game theory question to help in deconstructing situations where information quality is comprimised and requires valuation against a set of criteria. I would be interested to know any ...
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24 views

Congestion Game with Varying Price

I molded my problem as the following game (it is a congestion game with varying price): $N$ players share resources $E$, $S_i$ is the strategy space of player $i$ which is in $2^E$ (where $2^E$ is ...
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1answer
671 views

The core/Shapley value

Please help me to calculate the core of this easy coalitional game. I really didn't get it from my game theory course but want to understand the mechanism of calculating, describe it in detail please! ...