# Questions tagged [game-theory]

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].

2,192 questions
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### Game Theory, formulating a payoff matrix

Rowena and Colin play a hide-and-seek game. Rowena hides in one of 3 locations, and then Colin searches them in some order. If he searches in order i, j, k then his search cost is ci, ci + cj or ci + ...
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### Finding mixed strategy Nash Equilibrium in First Price Sealed Bid Auction with Complete Information

Let we think about $\textbf{First Price Sealed Bid Auction}$ in which bidder who offers highest bid wins the object. There are only two bidders in this auction for the simplicity, $I_1$ and $I_2$. ...
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### Can an open source project be considered a cooperative game?

Quote from https://opensource.google.com/docs/why/#engineering-economics: While open source work may have benevolent results, it is not an act of charity. Releasing work as open source and the ...
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### Reference for EV/ betting game questions. [on hold]

Need some good material to practice questions on probabilistic games. Mainly questions on the lines of finding EV and deciding whether it's a fair bet or not, or how much should one bet.
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### Week 1 Game Theory Example 7

Example 7. Verify that the strategies $y^{*}=(\frac{1}{2},\frac{1}{4},\frac{1}{4}),x^{*}=(\frac{1}{2},\frac{1}{4},\frac{1}{4})$ are optimal and $v=0$ is the value of the game $\overline{\Gamma}_{A}$ ...
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### Can a game theory strategy go in an endless loop?

Suppose there are two players in a game. Player A arrives at their best strategy by game theory but they also know that Player B knows about game theory, so Player B will also go with their best ...
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### joint venture capital determination (game theory)

In a joint venture project, n≥2 partners are to determine the amount of capital y to be invested in the company. They use the following rule. Simultaneously, each partner i submits a real number si≥0. ...
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### Is it possible to have one pure strategy and infinitely many mixed strategies?

I'm wondering if it is possible to construct a game in which there is one pure strategy and infinitely many mixed strategies? I don't believe this is true, since mixed strategies mix over pure ...
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### In game theory, what is the difference between garden of Eden and winning positions?

Conversely, what is the difference between traps and losing positions? Also, what is the relation between the set of winning position and garden of Eden? (I'm guessing the latter is the subset of the ...
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### Find the strategy to be left with legal moves

Two people are playing with $6$ piles of stones, $2$ piles each of similar type. The piles might be of different sizes. If a move consists of removing a particular type of pile completely and then ...
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### Nash Equilibrium Grandfather Problem

A grandfather is writing his will and must decide how much money he will leave to his three children: Sean, Tom, and Brad. The grandfather comes up with the following plan. Each of his children is ...
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### Altruistic rational agent

is there an agent in Game Theory (or even AGT), which goal is to minimizes other's costs? I've seen some agents that I would call "socialist", which goal is to minimize the sum, but including their ...
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### What is the Nash Equilibrium of the Dollar Auction?

The dollar auction is a type of auction in which players bid money for a dollar. Whoever bids the most pays what they bid and gets the dollar. Whoever bids the second most pays what they bid and get ...
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### Representing beauty contest with $n$ players in normal form game

I have a beauty contest question in which players must guess a number between $0$ and $5$. The closest score to ($p\times\text{average score}$) wins. Winners take 1 and losers take 0, whilst players ...
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### A game-theoretic chess puzzle — Proof verification

I recently came up with the following chess puzzle (which has almost nothing to do with one's Chess skills): Puzzle: Consider a variant of chess where black has to start with $1...e5$ regardless of ...
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### Nash-equilibrium in mixed strategies with arbitrary payoff function

So I came across this question in an old exam from the lecture I'm currently studying for regarding nash-equilibria in mixed strategies, which for the life of me I cannot seem to be able to find an ...
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### How are all kinds of equilibra in game theory connected to each other? (Please correct my summary)

I am a TA in a basic class on game theory in the economics department. I am ashamed to admit, but I do not entirely understand how all the different kinds of equilibria connected to each other. The ...
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### How to divide people into groups to maximize happiness

I need to split about 100 people into n number of groups: There should be equal numbers of freshmen, sophomores, juniors and seniors in each group. There are certain preferences, such as the ...
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### How to win in Battleship?

Battleship explained in wiki: (also Battleships or Sea Battle1) is a guessing game for two players. It is played on ruled grids (paper or board) on which each players fleet of ships (including ...
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### Derive formula for Shapley value

I have an exercise for Shapley value. As I know how to calculate it, I have an 8-element set and it would be difficult to calculate it by hand. It is a network G(V, E), where V is the set of nodes ...
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### Ex-ante expected utility in Bayesian games

I am reading the paper "Transition Models of Equilibrium Assessment in Bayesian Game" by Kiminao Kogiso, and I saw a quite new way to define the expected utility of a player in a Bayesian game. The ...
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### is there a winning strategy, and if so what is it?

Player 1 and player 2 play a game which consists of a rectangular grid with 3 rows and 20 columns. During each players turn they can colour a square in the grid (either a 1 x 1, 2 x 2 or a 3 x 3 ...
### Given $\epsilon \in [0, 1]$, find an analytic solution to $\underset{x \in \Delta_k | x_1 \ge \epsilon}{\text{argmax}}\;x^Tb$.
Let $\epsilon \in [0, 1]$, $b \in \mathbb R^k$, and $\Delta_k := \{x \in \mathbb R^k | x \ge 0,\; 1^Tx = 1\}$ be the unit $(k-1)$-dimensional simplex with $k\ge 2$. Question Find a closed-form ...
I'm trying to find the core of this cooperative game: $N = \{1,2,3\}$ and $v(\{1\})=24$, $v(\{2\})=24$, $v(\{3\})=26$, $v(\{1,2\})=42$, $v(\{1,3\})=44$, $v(\{N\})=52$. My solution: $x_1 \ge 24$...