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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21 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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1answer
17 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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0answers
10 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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2answers
261 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
18 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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0answers
3k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
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2answers
65 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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1answer
26 views
+50

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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1answer
3k views

Does Black have a winning strategy in Gomoku(freestyle)?

Gomoku is actually a finite two-person game of perfect information. Moreover, if we consider draw as victory of White, then by Zermelo's theorem, exactly one of the two has a winning strategy, either ...
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1answer
35 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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2answers
34 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: $$ ...
0
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1answer
55 views

Was my inference regarding $u(z)=log(z)$ correct?

I have already solved the problem but would appreciate a clarification in part (b). A has initial wealth $w$ and faces a loss $l$ with known probability $\pi$. Insurance available at unit price ...
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2answers
43 views

Need help with finding pure strategy nash equilbria

In the following game, how can I find the pure strategy Nash equilibria? The answers are apparently $(b,d)$ and $(b,g)$ but I'm not sure why. I have realised the following: Player one (rows) has ...
2
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2answers
57 views

Strategic form: Nash equilbrium

I am currently working through a question where I have to find any Nash equilibrium not in pure strategies, together with the associated payoffs. I have managed to identify the pure strategy Nash ...
2
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1answer
38 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 ...
0
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1answer
24 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 $y$, ...
0
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1answer
24 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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0answers
29 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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0answers
72 views

If $X$ is finite and $R$ is a complete and reflexive binary relation on $X$, then $M(R, S) \neq \emptyset$ on any $S \subset X$ iff $R$ is acyclic.

Could you help me to verify my proof and my writing? Definition 1: A binary relation $R$ on $X$ is complete if, for all $x, y \in X$ such that $x \neq y$,$xRy$ or $yRx$ or both and reflexive if, for ...
0
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2answers
95 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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2answers
37 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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2answers
61 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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2answers
44 views

if $G$ has no Nash equilibrium in pure strategies then $G$ has single Nash equilibrium in mixed strategies.

Let $G=(S,T,\pi _1 ,\pi_2)$ be a 2 player game with strategies $T$ for player 1 and $S$ for player 2 such that $|T|=|S|=2$, and payoff functions $\pi _1 ,\pi_2$. prove that if $G$ has no Nash ...
2
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1answer
80 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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0answers
39 views

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
478 views

Game Theory. Repeated Games. Strategy set.

I'm reading the book "Strategic games" by Krzysztof R. Apt. I have a question about the strategies in Prisoner Dilemma repeated game. On page 63 there is expression: "In the first round each player ...
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1answer
15 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 ...
0
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1answer
21 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 ...
0
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1answer
86 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 ...
2
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1answer
277 views

Solving a 3x3 payoff matrix

I need some help solving the value of this payoff matrix and finding the optimal strategy: $$ \begin{matrix} 1 & 2 & 4 \\ -1 & 5 & 3 \\ 3 & 3 & ...
0
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1answer
34 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 $v(S)=4$ ...
415
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25answers
53k views

Splitting a sandwich and not feeling deceived

This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an ...
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0answers
17 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 ...
0
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1answer
22 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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2answers
102 views

Game between Alice and Bob involving extremal numbers

Alice generates $4$ numbers in $(0,1)$ independently and uniformly at random. She discloses one of the numbers to Bob, who is requested to guess whether the disclosed number is extremal (i.e. the ...
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1answer
45 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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1answer
318 views

Reduce the payoff matrix using (weakly) dominated strategies

Below is the payoff matrix of a game. Use the principle of elimination of (weakly) dominated strategies to simplify the payoff matrix. What is the optimal solution of the game for the row player? ...
4
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1answer
499 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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1answer
32 views

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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0answers
16 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 ...
1
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1answer
50 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 ...
2
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0answers
75 views

A combinatorial game theory problem

In details, there are four bishops on a chessboard in two pairs. In each pair they sit in orthogonally adjacent squares. How many positions can there be to place the two pairs on the chessboard ...
1
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2answers
175 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 ...
4
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1answer
457 views

Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
77
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10answers
9k views

Mathematician vs. Computer: A Game

A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same ...
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1answer
44 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 ...
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1answer
880 views

Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?

It seems like most game theory tutorials focus on 2-player games and often algorithms for finding Nash equilibria break down with 3+ players. So here is a simple question: Is ...
2
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0answers
41 views

Correlation of belief distributions from distinct signals

Anne and Bob are two Bayesians who initially share a non-degenerate prior about a binary state of the world. Anne observes some signal (i.e., an experiment in Blackwell's terminology) about the state ...
6
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1answer
84 views

The game with countable amount of steps

Here is a cute problem. The angel and the devil play a game. Firstly the angel has an empty box and the devil has a box which contains all numbers from $\mathbb{N}$ (one copy of every natural ...
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0answers
28 views

Balancing a game, or proving the imbalance of a game

Consider a real-time game with two teams $T_1$ and $T_2$ fighting each other, each team composed of $n$ players $p_1^t,\dots,p_n^t$, where $p_i^t$ denote the $i$th player of the team $T_t$. Each ...