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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Calculating mixed strategy Nash equilibria: using the derivative?

From roaming around and looking for ways to calculate the mixed strategy Nash equilibrium, I learned that a general way to do it is by determining the probability of choosing a strategy in such a ...
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11 views

Static game with complete and incomplete information

I am currently trying to learn game theory on my own. I have a question regarding the solution methods for static games with complete information vs that of incomplete information. The textbook ...
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1answer
20 views

What is the optimal reserve price in a second price sealed bid auction?

Consider a seller who must sell a single private value good. There are two potential buyers, each with a valuation that can take on one of three values,θi∈{0,1,2}, each value occurring with an equal ...
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1answer
24 views

Revelation Principle

Would someone be so kind as to explain me the Revelation Principle with a simple example with two agents bidding for one good where one agent would lie about his perceived value of the good?
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1answer
41 views

Games on betting from a set

Two players each chooses a number from the set $\{1,2,4\}$ and correspondingly bets an amount of \$$1$, \$$2$, or \$$4$. There is no collaboration between players. Rules: $1.$ If the two chosen ...
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1answer
30 views

A 2 Player Pure Strategy Game

There are two players each has $n$ balls. At the same time they distribute their balls among $m$ boxes. For each box 1 point is given to the player with more balls and zero points to other one (When a ...
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50 views

How can I tell if a two-person game is non-degenerate, given its payoff matrices?

Consider a two-person game with payoff matrices defined by \begin{equation} P= \left( \begin{array}{ccc} 0 & 4 & 1 \\ 2 & 2 & 4 \\ 3 & 2 & 2 \end{array} \right) \quad ...
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1answer
47 views

Payoff matrix with a specific form

I am very stuck on this question: Suppose that $b \in \mathbb{R}^m$, $c \in \mathbb{R}^n$, $A$ is a $m \times n$ real matrix, and all components of $A$, $b$ and $c$ are positive. Consider the ...
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3answers
72 views

Is there a game theory for doing the dishes in a shared living situation?

It occurred to me this morning (when I was intentionally not tidying up my flatmate's dishes) that doing the dishes in a shared living situation, such as at an office, or living with housemates, might ...
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29 views

Optimization problem with variables in the subscript

I want to solve a optimization problem, which mimics the actions between a seller and several buyers. A seller has several goods, 1, 2, ... J, with prices $p_j$ and quantity $q_j$. A buyer can only ...
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1answer
18 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 no ...
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1answer
39 views

Should you choose highly owned or little owned players in fantasy sport

Here's the situation: It's a fantasy soccer game where players score points for my team based on their actual performances on the pitch. I have a team of 11 players and their is no limit to the ...
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1answer
41 views

Multiplayer finitely ultimatum game

Imagine a 3 member legislature that must decide how to allocate an asset of unit value. There are three rounds to the game and in each round a randomly assigned proposer must make an offer to each of ...
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1answer
27 views

Could there be multiple symmetric equilibriums in a symmetric games?

Given a finite symmetric 2 player game with a strategy space $S$, a (mixed-strategy) symmetric equilibrium is a distribution $d\in \Delta(S)$ such that $(d,d)$ is a Nash equilibrium. A known result ...
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12 views

Is there a name for the ratio between the optimal social-welfare equilibrium and the worst social-welfare equilibrium of a strategic game?

Suppose you have a $n$ players strategic game, and assume that the "social-welfare"(SW) of the game is defined as the sum of payoffs to the players. Two well known measures about the "efficiency" of ...
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1answer
26 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 ...
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21 views

What is the Hamel or Schauder basis for functions from the subsets of the natural numbers to the reals

I'm trying to prove that some linear operator (Shapley value) $\varphi:\mathbb{R}^{P(\mathbb{N})}\rightarrow\mathbb{R}^{\aleph_0}$ is unique, where I'm using $\mathbb{R}^{P(\mathbb{N})}$ to denote all ...
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1answer
38 views

convex for nash equilibrium

I have trouble understanding this question, the first question to my understanding is asking me that for a fixed p , (p,q) is nash equilibrium, prove that all (p,q) are convex. and for the ...
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1answer
25 views

What is the pareto optimal payoff vectors for war of attrition game?

The game works as follows: two player are involved in a dispute over an item. the value of the object to player i is vi>0. time is modeled as a continuous variable that starts at 0 and runs ...
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1answer
27 views

how to find mixed Nash equilibria for 3x3?

A (3,2)(3,0)(2,2) B (1,0)(3,3)(0,3) C (0,2)(0,0)(3,2) p q 1-p-q So what I have done is : 3p+3q+2(1-p-q)=p+3q q=1 this is when A=B p+3q=3(1-p-q) p=-3/4 this is when B=c I don't know ...
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2answers
25 views

prove that if $n=k$ then white has a winning strategy in $S_{n,k}$.

Black and white play sequentially the game $S_{n,k}$ with $k,n\in \mathbb N \space 0\leq k\leq n$ the game board consists of all subsets $A\subseteq\{1,2,...,n\}$ such that $1\leq |A|\leq k$. every ...
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71 views

Can “tit for tat” strategy be defined in monadic second-order logic?

Prisoner's dilema game can be represented as a game tree, which could be infinite game with corresponding infinite game (binary) tree in common case. There is well-known tit for tat strategy, which ...
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Optimal Strategy Game of Communicating without Overlap

Today, I had a conversation which proceeded very poorly; indeed, I had this conversation with $n$ people, including myself, and everyone had something to say. Problem was that, for an unreasonably ...
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1answer
29 views

Probability of coin flip betting

Imagine a situation where you and a friend both have 5 dollars, and you play him in a 50/50 coin flip "duel" where if it flips heads you receive a dollar from them otherwise you lose a dollar to the ...
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1answer
13 views

Difference in the function (Game Theory)

I was hoping to know the difference between equation (1) and equation (2). Would they be considered equal or is equation (2) less strict compared with equation (1)? Let the correspondence ...
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1answer
17 views

What is the difference of pure strategic equilibria and nash equilibria?

Are they the same thing just named differently or with minor differences?
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1answer
25 views

What is a pareto optimal strategy for two player general sum games?

What is the definition that a strategy for player one and player two to be pareto optimal?
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1answer
40 views

Utility index function-would appreciate some clarification (confused about a log function)

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 p will ...
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1answer
72 views

Nash's Axiomatic Bargaining: Source of problems sets and practice questions.

From where can I practice questions related to the following topic: Nash's Axiomatic Bargaining. Any form of book reference or a link to some online problem set would be highly appreciated.
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1answer
39 views

Prove two person gerneral sum game , the expected payoff cannot be lower than safety level.

Prove that in a two-person general sum game, the expected payoff of any player at any Strategic Equilibrium (mixed or pure) can not be smaller than the safety level of this player. How do I prove ...
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3answers
139 views

Prove using a strategy stealing argument that player 1 has a winning strategy in the chomp game

I have no idea what this question is asking or how to prove it mathematically. I realize based on the strategy stealing theory that if player two has a winning stratagy then player one can use the ...
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1answer
16 views

Determining the Likelihoods of Different Game States

Suppose a game is played in which Player 1 must gain two points to win and Player 2 must gain five points to win. Both players start with zero points. In any round, Player 1 has a $1/3$ chance of ...
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62 views

A tough dice probability conundrum

There are 2 fair die. You randomly roll them at the same time. To win is to have the sum of the die or some combination of the sum or the value of each die cover the numbers 1-9 inclusive. You can ...
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1answer
43 views

Game Theory Nash equilibrium

How do we do in order to find Nash Equilibrium in a $3$ players game with $3$ possible strategies for each player?
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1answer
21 views

Proof sprague-grundy value is 0 if and only if it is losing position

So, i take this game theory module this summer, and i encountered this exercise problem, i tried to do this by induction by have terminal position (grundy-value = 0) as base case, but can't figure out ...
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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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Bingo based on number of hits on card?

I was wondering how many possible combinations there were to win based on the number of hits on a bingo card (25 spots, 1-75 etc.)? I know that if you get 1,2,3 hits on the card, there are no chances ...
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Legendre transform and Minimax Theorems.

Denote the class of lower-semi-continuous convex functions $f:\mathbb{R}^n\to \mathbb{R}\cup\{\pm\infty\}$ by $Lscx(\mathbb{R}^n)$ ( so that only function attaining the value $-\infty$ is the constant ...
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1answer
46 views

Formulating recurrence relation

Alice and Bob worked in a restaurant and received n currency notes in total as tips. Every note has a value (either 1 dollar, 5 dollar or 10 dollar) written on it. The currency notes are arranged from ...
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1answer
85 views

Axiomatic Bargaining: Nash's Solution

The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc. Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:. Two individuals can ...
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1answer
32 views

Bayesian update from uniform prior to uniform posterior ?!?

I was working through a signaling game problem recently and the proof suggested the following: Actor A has a type: $\ \mathscr{t} \sim Uniform[-1,1]$ Actor A gives signal $\pi^*$ that perfectly ...
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28 views

Discrete Structrue

I was stuck with the following problem. Two players A and B play a game where they take turns adding numbers from 1 through 10, and the first person who gets to the target of 100 wins. Assume A ...
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2answers
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Best strategy for this game. [Nintendo Wii game]

Which is the best strategy for this game? Actually is a Nintendo Wii game. It's a 4 people game. There is a ladder of 10 steps. Each player says a number (they can chose between those numbers ...
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1answer
24 views

The strategy of row player in $2\times 2$ ordinal game

I am proving the simplest part of theory of moves. Assume that both players are rational, and consider the following $2\times 2$ ordinal game: $$\begin{bmatrix} (3,3) && (2,4) \\ (4,2) ...
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1answer
54 views

Why is the “rational” solution to the Traveler's Dilemma 2?

Traveler's Dilemma An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical items. An airline manager tasked to settle the ...
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Approximate Solution to Backwards Recurrence of Dynamic Game

Suppose we keep tossing a fair dice until we reach some cumulative sum greater than or equal to $N$. Then, let $S_k$ be the expected value of the final sum, given that the current sum is $k$. We have ...
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Expected value for number

Suppose you have a game with $n$ stages. For every stage $i$, you have $p(i)$ probability to advance to the next stage, and $1-p(i)$ probability to return to stage $1$. You win the game by advancing ...
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1answer
76 views

Marriage Market Proof (Alternative Proof of Rural Hospitals Theorem)

How do I get (a) + (b) + (c) $\implies$ (d) $\implies$ (e)? (a) Show that for each $m \in M$, if $\mu(m) = \emptyset$ for some stable matching $\mu$, then for the woman-optimal matching, $\mu_W$, ...
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Repeated games with observable actions

I'm going to hold a lecture in a study circle about Repeated games and observable actions and also a little about Repeated games with imperfect public information. We are following the book of ...
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1answer
78 views

Toss a fair die until the cumulative sum is a perfect square-Expected Value

Suppose we keep tossing a fair dice until we want to stop, at which point the game ends and our score is the cumulative sum, or until the cumulative sum is a perfect square, in which case we lose and ...