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

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2answers
209 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 ...
4
votes
1answer
267 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. $k$ ...
0
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1answer
154 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 ...
2
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1answer
151 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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0answers
88 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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1answer
183 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 ...
0
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1answer
122 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
5k 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, ...
4
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0answers
149 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 ...
7
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4answers
148 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
78 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
3k 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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1answer
72 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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0answers
65 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
700 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} ...
2
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1answer
138 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
802 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 ...
0
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1answer
54 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
610 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 ...
5
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2answers
475 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 ...
0
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1answer
231 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.
4
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1answer
848 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 ...
0
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1answer
546 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
115 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
165 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 ...
0
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1answer
274 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 & ...
0
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1answer
106 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
117 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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0answers
92 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 ...
0
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0answers
27 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 ...
0
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1answer
1k 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! ...
6
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2answers
275 views

How to solve this puzzle?

There are $N$ consecutive doors. Two players 'B' and 'J' plays a game. Both take turns alternately, and in each turn a player can open any one door. They define a block of 3 consecutive open doors as ...
6
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2answers
571 views

What is the Nash Equilibrium of the Monty Hall Problem?

The Monty Hall problem or paradox is famous and well-studied. But what confused me about the description was an unstated assumption. Suppose you're on a game show, and you're given the choice of ...
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1answer
233 views

Computability of busy-beaver sequence? [closed]

We can draw a parallel between cellular automata and busy-beaver numbers. For example the initial case occupies some kxk square in the plane,leaving all the other cells emty, after how many ...
2
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1answer
56 views

Cardinality of strategy space of $G_{\omega}(\mathbb{R})$ up to an equivalence relation

Suppose, in $G_{\omega}(\mathbb{R})$, a player's two strategies are equivalent, if, for any strategy of his opponent, the outcome incurred are the same. It can be shown that in $G_{\omega}(\omega)$ ...
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2answers
637 views

common knowledge and concept of coarsening partition

Here is a proof of the equivalence between my definition and Aumann's for "common knowledge". I'm assuming some familiarity with set partitions. Aumann's definition is in terms of the ...
2
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0answers
198 views

What is the (expected) outcome of this hybrid auction?

A certain hybrid auction can be accurately modelled as follows. There are $n$ risk-neutral, rational participants $i=1,2,\ldots,n$, and a guy called Zerro: $i=0$. Each, except Zerro, has a private ...
9
votes
5answers
293 views

How can the observed strategies* in this actual auction be explained?

This is a "real world" question. Recently I witnessed the separate auctions of about 30 houses. The place where I went uses the following rules. The following describes the procedure for the ...
1
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1answer
115 views

$\subset$-minimal set to determine an opponent's strategies up to an equivalence relation

In a two-player sequential game, each player's strategy is a function that maps history records to her own action space. Naturally, once two players' strategies are given, the outcome, a vector of ...
5
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2answers
695 views

Game theory Computing pure Nash equilibrium probability

We have a $2$-player game and each player has $n$ strategies. The payoffs for each player are in range $\left[0,1\right]$ and are selected at random. Show that the probability that this random game ...
1
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1answer
62 views

Probability of winning of teacher

Teacher is playing a game with his students. He is having $k$ red balls. Each of his student is either having a red or black ball. $M$ students have red balls and $N$ students have black balls. Now ...
1
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1answer
804 views

Understanding common knowledge in logic and game theory

For $k = 2$, it is merely "first-order" knowledge. Each blue-eyed person knows that there is someone with blue eyes, but each blue eyed person does ''not'' know that the other blue-eyed person ...
3
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2answers
56 views

Obtaining certain pairs of numbers using three “machines”

Each of three machines can read a card on which is written a pair of whole numbers $(m,n)$ and print a new card. Machine $\text{A}$ reads $(m,n)$ and prints $(m-n,n)$. Machine $\text{B}$ ...
2
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0answers
150 views

Comparing Nash equilibrium and Pareto optimal actions

Suppose that $(x_{i}, x_{j})$ identify actions for two players $(i,j)$. If we define Pareto optimal actions by $$h(x_i) +h(x_j)- \eta[p(x_i)+p(x_j)]=2\gamma$$ and Nash equilibrium actions by ...
2
votes
1answer
248 views

Game Theory: determining the value of the foxhole game

"A soldier can hide in one of five foxholes, and a gunner can hide in four spots: A, B C, and D. The configuration looks like this: 1 (A) 2 (B) 3 (C) 4 (D) 5. If a shot is fired at a location and the ...
0
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2answers
72 views

Game theory strategy equilibrium that concerns all players' strategies, not just himself

Nash equilibrium occurs when there is no benefit gained by changing its strategy unilaterally from the equilibrium strategy. So is there any equilibrium named for the following case: when there is no ...
6
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1answer
1k views

Finding mixed Nash equilibria in continuous games

I'm taking my first (graduate-level) game theory class. I understand how to find Nash equilibria in simple games, such as those given in finite tables, and can see (usually) how to find the mixed ...
2
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3answers
59 views

How to maximize chances to win?

Let's say I have $n$ stacks of coins such as the $i$-th stack contains $c_i$ coins at the beginning of a game. The game is simple. The players have to take any number of coins from one stack (and ...
2
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0answers
240 views

A dynamic Stackelberg game - general characterization

my question is about general representation of a dynamic Stackelberg game which is played in continuous time. We have maximization problems of two agents who play this game. Agents are 'Leader' and ...
0
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1answer
385 views

Bondareva-Shapley theorem

By definition, an imputation is a vector $\alpha \in \Re^n$ such that (1) $~ \alpha_i \ge v(i)~~~\forall i \in N $ (2) $ \sum_{i \in N} \alpha_i = v(N) $ where N is the coalition with all the ...