Questions tagged [nash-equilibrium]

For questions regarding the the Nash equilibrium solution concept in strategic games.

Let $(S,f)$ be a game with $n$ players, where $S_{i}$ is the strategy set for player $i$, $S=S_{1}\times S_{2}\times \ldots \times S_{n}$ is the set of strategy profiles and $f(x)=(f_{1}(x),\dotsc ,f_{n}(x))$ is its payoff function evaluated at $x\in S$.

Let $x_{i}$ be a strategy profile of player $i$,$x_{-i}$ be a strategy profile of all players except for player $i$. When each player $i\in \{1,\dotsc ,n\}$ chooses strategy $x_{i}$ resulting in strategy profile $x=(x_{1},\dotsc ,x_{n})$ then player $i$ obtains payoff $f_{i}(x)$. Note that the payoff depends on the strategy profile chosen, i.e., on the strategy chosen by player $i$ as well as the strategies chosen by all the other players. A strategy profile $x^{*}\in S$ is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is

$$\forall i, x_i \in S_i: f_i(x_i^{*}, x_{-i}^{*}) \ge f_i(x_i, x_{-i}^{*}).$$

532 questions
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How to compute ALL Nash equilibria in an example of a 3x3 matrix

I am trying to understand how to compute all Nash equilibria in a 2 player game, but I fail when there are more than 2 possible options to play. Could somebody explain to me how to calculate a matrix like this (without computer) \begin{matrix} …
user106371
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best strategies for 'Squid Game' episode 6 game 4 "marble game" players

Two players each get $n=10$ marbles. Every alternating turn, one player (say, the player whose turn it is) hides an amount of own marbles in fist, and, the other player must guess if hidden amount is odd or even, and, that other player (i.e., the…
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Optimal strategy for Jackpot Rock Paper Scissors

Jackpot Rock Paper Scissors is a gambling variant of Rock Paper Scissors, wherein ties result in the wager being carried forward into a jackpot. If a player plays the same hand (rock, paper or scissors) 3 or more consecutive turns and wins on that…
Heptic
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Equilibrium existence proof

Problem: Let $J$ be an integer and let $I$ be an integer multiple of $J$. Let ${\cal I}= \lbrace 1,2,\ldots, I\rbrace$ and ${\cal J}= \lbrace 1,2,\ldots, J\rbrace$. The set $X_{I,J}$ of all $\frac{I}{J}$-subsets of $\cal I$ contains exactly…
Shane
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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 numbers do not have to be in whole dollar units. The…
Ray B.
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Optimal Strategy for Rock Paper Scissors with different rewards

Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward. If you win with Rock, you get \$9. Your opponent loses the \$9. If you win with Paper, you get \$3. Your opponent loses the \$3. If you win with…
Heptic
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Is there experimental evidence that people ever play mixed Nash equilibrium in real games?

Have any studies been done that demonstrate people (not game theorists) actually using mixed Nash equilibrium as their strategy in a game?
logicbird
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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 equilibria in those cases. However, most of our…
Hanmyo
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Blotto game variation

My smart friend ZWX challenged me to solve the "brainteaser" below, but to my surprise, the problem seems highly nontrivial as I took a closer look. Anyway, the question goes: In a game, both you and your opponent have $100$ units of resources to…
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Can Nash Equilibrium be more than two?

In the Prisoner's Dilemma example, we know that there is only one Nash Equilibrium. That is both of them confess. Is it possible that there are two Nash equilibriums in one example? Can you roughly give me such an example?
sleeve chen
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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 three doors: behind one door is a car; behind the…
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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 $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ the only Nash equilibrium in a…
Wesley Tansey
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Nash equilibrium in first price auction

I'm trying to understand Exercise 18.2 from Martin J. Osborne and Ariel Rubinstein A Course in Game Theory about finding pure Nash equilibria in a first-price auction. There are $n$ players, named $\{1,2,\dots,n\}$ participating in a sealed-bid…
Mirzhan Irkegulov
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Pure vs mixed strategy Nash Equilibria

Just learning about Nash Equilibria. The pure strategy one is explained as an outcome where both/all players feel like they couldn't have done better given what the others were doing. Mixed strategy is one where even after announcing your strategy…
Balazs Rau
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Find the Nash equilibria for a joint project modeled as a strategic game

I am currently working out of the Introduction to Game theory book by Martin J Osborne and I am looking to solve this given question: Two people are engaged in a joint project. If each person $i$ puts in the effort $x_i$, a nonnegative number…
user481610
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