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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Characterization of pre-orders

Let $X$ be an arbitrary set, and $\leq$ be a pre-order on $X$. Does there always exist $u:X\to \Bbb R$ such that $x'\leq x''$ iff $u(x') \leq u(x'')$? If this is not true in general, is that true ...
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40 views

Clarification of Game Theory Exercise

Look at the second exercise here: (open yale course on game theory) http://oyc.yale.edu/sites/default/files/problemset1_1.pdf Does anyone understand the meaning of this from question a) "What ...
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1answer
49 views

What is the total number of possible chess moves at White's second turn?

At White's first turn there are $20$ possible moves: each pawn can move forward one or two spaces, or a knight can jump over the pawns to one of two positions each. Some moves are of course likelier ...
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168 views

Lemke-Howson pivoting in degenerate bimatrix games

I'm working on an implementation of the Lemke-Howson algorithm for finding Mixed Nash Equilibria of bimatrix games, and I'm running into a roadblock when the algorithm is fed a degenerate game. For ...
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29 views

Coalition games

Let us define $\gamma([n],v)$ and $(\gamma'([n],v')$ as the two cooperative games in coalition form. Both games have the same set of players. Let this hold for every non-empty coalition $S$: $v(S) ...
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Can one player win more games while scoring fewer points over multiple trials a simple probabilistic game?

The Game Two players $P_{1}$ and $P_{2}$ will play the following game: Three non-negative numbers $a$, $b$, and $c$ will be selected at random from a distribution unknown to either player. The ...
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40 views

Expected revenue in first-price auction with budget constraint drawn uniformly between [0,1]

I am trying to understand an example from the article "Standard Auctions with Financially Constrained Bidders" Che & Gale (1998) - Review of Economic Studies. Two buyers each value an object at ...
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7 views

Where can i find good tutorials of the simplex algorithm,hungarian algorithm and lagrange optimization?

Where can i find good tutorials of the simplex algorithm,hungarian algorithm and lagrange optimization? My textbook is kind of vague and i would love to learn these fascinating algorithms as good as ...
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42 views

How do I show that this game played on a Markov chain has a unique Nash equilibrium?

There are $k$ stages in this game, and each stage is worth one unit of utility to a player (of which there are $n$). Each player $i$ finishes stages at a rate $\lambda_i$ (in a continuous time Markov ...
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65 views

Topology and Differential Games

I'm a engineer who is making research on differential games in multiagent control. I was reading a tutorial on differential games and the author advised to get the required math background from the ...
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117 views

Splitting the dollar Nash equilibrium

I'm working on a game theory problem I can't seem to figure out. Players 1 and 2 are bargaining over how to split $\$10$. Each player names an amount $s_i$, between 0 and 10 for herself. These ...
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19 views

Sub-game perfection when an agent is indifferent

If looking at one of the sub-games the player is indifferent between two actions. How does the backward induction work to recognize sub-game perfection? I.e., suppose player 3 has two action $A_3 = ...
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57 views

Winning or losing in chess - a question of combinatorics?

I have observed that every chess game can be assumed as a sequence of moves that lead either to win or to lose (in a few cases to a drawn game). It is very interesting to Count all the moves that are ...
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173 views

stable marriage algorithm problem

Better of the two Suppose that in the stable marriage problem with $n$ men and $n$ women, we have found two (possibly different) stable matchings $S$ and $T$. We will show how to combine $S$ and $T$ ...
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42 views

Best strategy for this archery-based probability game

This is with reference to the comments posted by @Trenin on my answer to this question. He says that since 2 players strategies depend on each other, we can't get the best strategy so easily. My ...
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1answer
31 views

Mixed Nash equilibrium in two players game three stategies

I have this problem about finding the mixed Nash equilibrium. The payoff matrix is the following A(p) B(q) C(1-p-q) A 4 0 0 B 0 4 0 C ...
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23 views

Does this payoff matrix correlate to the problem statement for the game?

The problem statement: Consider a two agent game in which the possible action for agents are C and D. The utilities and therefore preferences for agent $i$ among strategy choices are: $$ ...
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72 views

Game Theory: Variants of the Stag Hunt

I have briefly answered question a but I am stuck on question b and am not sure how to go about answering it. If anyone could help it would be greatly appreciated. Thanks in advance
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38 views

Game Theory - Contributing to a public good

I have attempted to answer the question but I think I am trying to answer it in a very difficult way as the algebra gets messy and confusing. If anyone could help me out it would be greatly ...
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1answer
57 views

Game Theory - n-player friends/enemies game

"Consider the following game with n players $\{1, . . . , n\}$. Each player is invited to two parties. All players like to party, but the parties are on the same day, so each player has to decide ...
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Showing $\frac{1+c}{a+b}\leq \varphi$, when $\frac{1}{1+b}\leq a\leq 1$, $\frac{c^2}{a+c}\leq b\leq 1$ and $0\leq c\leq 1$

The title pretty much says it all. Let $\varphi\triangleq\frac{1+\sqrt 5}{2}$ be the golden ratio. Let $a,b,c$ be some non-negative numbers such that: $\frac{1}{1+b}\leq a\leq 1$ ...
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1answer
29 views

Weakly acyclic games in game theory

I read that weakly acyclic games are more general than potential games. Potential games are said to have a finite improvement property where each player's payoff function is aligned with a potential ...
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173 views

Dominant-Strategy Equilibrium vs Nash Equilibrium

What's the difference between dominant-strategy solution and Nash Equilibrium? I could not tell the difference judging from the definitions. It would be appreciated if these concepts can be ...
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34 views

Who's winning this coin drawing game?

There are 2 piles of coins, each containing 2010 pieces. Two players A and B play a game taking turns (A plays first). In each turn the player on play has to take 1 or more coins from 1 pile or ...
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69 views

Pareto optimality - Game theory

Good morning, I have this game theory problem. Let's consider 5 farmers, each of them has 2 cows to put into the field. So every farmers can put 0,1 or 2 cows. I denote the three stategies by ...
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52 views

Single bid auction: calculating bid as function of winning probability

I'm simulating a auction game with sealed single bid, where each of the $n$ players has winning probability $p_i,i=1,...,n$, and their bids $b_i$ have to be calculated to meet the $p_i$. Supposing ...
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37 views

Reducing an I-optimal problem to a Pareto-optimal problem

Given a set $\textbf y\subset\mathbb R^2$, let $y = (y_1,y_2), y'=(y'_1,y'_2)\in\textbf y$ be elements of that set, let $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...
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91 views

Nash equilibrium: Can I delete weakly dominated strategies in this case?

As far as I know, an equilibrium can involve a weakly dominated strategy, but cannot involve a strictly dominated strategy. Is there a general rule for when/if you can safely delete a weakly dominated ...
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31 views

Biobjective optimisation, pareto non-domination

Ok, so, I have a function $f_I(y_1, y_2) = \max\{\alpha y_1 + (1-\alpha)y_2:\alpha\in[\alpha_{min},\alpha_{max}]\}$ that I'm trying to minimise, and I'm asked to find, amongst a set of vectors $y$, ...
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1answer
71 views

Calculating Nash equilibrium in mixed strategy in a game where a Nash equilibrium in pure strategy exists

Let's say I want to calculate Nash equilibrium with mixed strategies for a two-players game, in which there is no Nash equilibrium with pure strategies (no dominant strategy for any of the two ...
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2answers
76 views

Street Fighter: is the game balanced?

Suppose that $A$ is a matrix that describes the matchup information of any pair of Street Fighter characters e.g., considering $3$ characters, assume that the first row/collumn is associated with a ...
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47 views

optimal strategies for 2-player zero-sum games of perfect information

Do finite-state 2-player zero-sum games of perfect information with only win-draw-loss outcomes always have deterministic-and-memoryless optimal strategies for both players? In other words, ...
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Book or paper recommendation about “Rube Goldberg Mathematics” // e.g. Longest path problems

First: My question is not be very specific, since I lack a concrete overview, but my idea/thoughts in a nutshell: I would like to have a recommendation of a good book, paper or article about processes ...
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1answer
67 views

Cournot Duopoly Game - Nash equilibrium

I have this problem about Cournot Duopoly game. Actually I don't know if I have understood the "real sense" of the problem. I consider CD game described by the following payoff fucntions: $$ ...
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84 views

Math theories in Game Theory

What are all the mathematical theories in Game Theory? I have taken Mathematical Modelling, including: application of linear systems, matrix operations, inverse of matrix, leontif input-output model, ...
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133 views

Finding the Nucleolus

Given the following table of values and excesses of coalitions S and imputation $\vec{x} = (9,6,9)$: How do I find the Nucleolus? My book wasnt clear on the method of calculating it, so Id like to ...
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1answer
25 views

Understanding the proof of Gibbard-Satterthwaite theorem

Let $n$ be the number of voters and $A$ be the set of alternatives. For voter $i$, we denote by $a \succ_i b$, if $i$ prefers $a$ to $b$, where $a,b \in A$. Let $L(A)$ denote the set of all strict ...
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67 views

Finding support of optimal mixed strategy

Considering a finite zero-sum game represented by a matrix $\mathbf{A}$, I understand one method to solve for the game is to use the principle of indifference. If the optimal strategy of Player 1 (who ...
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87 views

Implicit function theorem in comparative static problem

The individual lives for two periods. He has a utility function $u(c_{1} )+ bu(c_2)$. His budget constraint requires that his period I consumption be his period I endowment minus any savings, $c_1 = ...
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1answer
42 views

Determining the core of a permutation game

For my game theory class I have to determine the core of the following cooperative game: $\mathrm{N}={1,2,3}$ $S=\{1\}$ gives $v(S)=2$ $S=\{2\}$ gives $v(S)=5$ $S=\{3\}$ gives $v(S)=4$ ...
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24 views

Translating these “payoff matrices” to normal form.

I'm used to looking at payoffs to given players in normal form but here I'm given the following matrices and asked to interpret them as payoff matrices. Can someone explain how I'd translate the given ...
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1answer
34 views

Optimum type auction for seller's profit maximisation

I am trying to formulate an auction, in which sellers will create a cartel and ask for the highest possible price that buyers pay. Practically, I would have a multi-part game in which in each part I ...
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1answer
122 views

Finding the nucleolus of a jury case

I am a bit stuck on this problem: (a) $\quad $ Consider a jury system with $12$ jurors in which a defendant is found guilty if voted guilty by $10$ or more of the jurors. We represent this jury ...
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1answer
75 views

Game theory on selling shoes

I am stuck in this question: A pair of shoes consists of a left shoe and a right shoe, and can be sold together for $ \$10 $. Consider a coalitional game with $a+b$ players: $a$ of the players have ...
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19 views

every simple, monotone 3 player TU game is a weighted majority game.

I'm trying to prove that every 3 player TU game $G=(\{1,2,3\},v)$ which satisfies: $G$ is simple: if $ T\subseteq N$ then $ v(T)\in \{0,1 \}$ $G$ is monotone: if $T\subseteq S $ then $ v(T)=1 ...
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What is the optimal strategy for a peace war game with unequal power and varying peace agreements?

Consider a variant of the peace war game in which nation A can "harm" nation B much more than B can harm A if they both go to war, but each nation can also give the other nation tribute. To formalize ...
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129 views

Problem in game theory related to traffic networks

I have learnt game theory for a short period of time and I am not familiar with multi-player non-zero sum games. Here is a problem from my book which I am stuck: In this road network below each of ...
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1answer
67 views

Winning strategy for solitaire?

I'm talking about the Klondike solitaire, turning three cards at once to the waste and placing no limit on passes through the deck. I know there isn't always a winning strategy, a counterexample can ...
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248 views

Alice and Bob number sum game

Alice and Bob play a game with first $N$ positive numbers. Out of these $N$ integers some $K$ integers are missing. So both decided to play with remaining $N-K$ integers and in this game Alice wants ...
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62 views

Extensive and strategic form in game with uncertainty

I have to solve the following problem for my gametheory course: Software Inc. and Hardware Inc. are in a joint venture together. The parts used can be defective or not; the probability of defective ...