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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Why does the 1st player in this subset take-away game always have a winning strategy?

This is a HW problem of mine that I cannot, for the life of me, figure out. There is a take-away game where there are a number of elements A, and the person that wins is that last person to remove a ...
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1answer
30 views

Nash equilibrium for two players game.

Consider a game for two players, say "Player A" and "Player B". The two sets of strategies are denoted by $A$ and $B$, available to the players. Consider a symmetric situation where the players have ...
8
votes
1answer
333 views

Expected travel of random walk in arbitrary game with multiple payouts

As explained here, the average distance or 'travel' of a random walk with $N$ coin tosses approaches: $$\sqrt{\dfrac{2N}{\pi}}$$ What a beautiful result - who would've thought $\pi$ was involved! ...
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1answer
38 views

When solving a system of equations for a game theory question, can the solutions be negative?

I have a homework question on solving a game matrix geometrically. $m =$ $\begin{bmatrix}1 & 11\\7 & 2\end{bmatrix}$ (after adding the constant $k$ to ensure it's a positive matrix) The ...
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1answer
65 views

Find a Nash equilibrium solver

The solvers I know so far are designed only to allow payoffs as given numbers. But is there a solver allowing users to type payoffs as variables?
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0answers
85 views

Finding Nash equilibrium in game with random event at event tree?

I have posted a question about finding the NE of sequential game with imperfect information. It is lucky that the game can solve could be dealt with could be dealt with by a simpler argument. Here is ...
2
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0answers
100 views

A fashion victim puzzle

Consider $n \in \mathbb{N}$ fashion-sensitive kids, each wearing a T-shirt; for simplicity, kid $i \in \{1, \ldots, n\}$ initially wears shirt $i$. Tastes over the shirts are summarized in an $n ...
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votes
1answer
70 views

How to find the Nash equilibrium or subgame perfect equilibirum in a sequential game with imperfect information? [closed]

I have a problem with the sequential game with random event at the event tree. The model of the game as follows: Player = $\{A,B\}$ Pure strategy of player $A: A1, A2, A3$ For each strategy of ...
0
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1answer
166 views

IESDS and Nash Equilibrium - same solution [closed]

Applying the Iterated Elimination of Strictly Dominated Strategies (IESDS) to a game resulted with the same solution of the Nash Equilibrium. What does this imply? Actually that specific "quadrant" ...
0
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1answer
114 views

English translation of von Neumann's “Zur Theorie der Gesellschaftsspiele”, 1928

Some colleagues and I are reading various classic papers. We would like to read von Neumann's "Zur Theorie der Gesellschaftsspiele", 1928, but do not read German. Do you know of an English ...
4
votes
2answers
84 views

System of Differential Equations- Asymmetric First-Price Auction

I am working on a problem in my Auction Theory textbook regarding a two-player asymmetric first price auction. Assume the bidders are risk neutral. The problem statement is as follows: ...
5
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0answers
192 views

Cutting a Banach-Tarski Cake

I was reading a cake-cutting problem here (not really related, so I won't link to it), and for some reason, this variation occurred to me. I have no idea whether this problem is even well-formed: ...
3
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1answer
82 views

A deadly game of two werewolves and two townsfolk

This question was closed due to lack of own effort shown. Because I like the game of werewolf (a.k.a. Mafia) and thought it was a nice idea to pose a simplified version of it as a game-theoretic ...
0
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1answer
58 views

What is the mixed strategy equilibrium bid, if any, for complete information auction games with minimum bid?

Consider the following complete-information, auction game. There are two players $i=1,2$. Each bids simultaneously a value $b_i\in[0,\infty)$. The payoff function is symmetric: $$ \pi_i ...
2
votes
1answer
112 views

Mixed strategy problem - game theory

I have a basic doubt in a question of game theory. Assume that in a $2$ player game the mixed strategy profile $((a,b,0),(c,d,0))$ is a mixed strategy NE. Does the indifference condition in a mixed ...
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0answers
23 views

super-additive, sub-additive, and shapely value limitations?

I am working on the coalition formation. Most of the scientist used concept of shapely value for distributing the utility among the members of coalition. Up to my understanding, shapely value is good ...
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2answers
595 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
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0answers
43 views

Expectation of a continuous function

Can someone help with the following? I have a continuous function $g: A_i \times A_{-i} \to \mathbb{R}^k$, and a probability measure $\mu \in \Delta(A_{-i})$. We can let $A_i=\mathbb{R}^n$ and ...
0
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1answer
80 views

Is this a game theory problem or optimization problem?

Consider a problem that looks for a $x$ that can make the following problem into some equilibrium state (similar to an equilibrium solution to a min-max problem in game theory) $$ \max_x f(x)$$ $$ ...
0
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1answer
24 views

imperfect information in extensive form

Hi I'm trying to understand how to convert into extensive form this imperfect information game. consider the second graph of this example taken from example of imperfect information game in extensive ...
1
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1answer
59 views

Is my cake split envy-free (and coalition-resistant)?

I once read that splitting a cake in 4 parts envy-free is notoriouse difficult. Not to mention splitting it with 5 or more people. Methods involve arbitrarily long recursions and cake split onto ...
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0answers
37 views

Example of infinite game without any Nash equiblibria

I have to find an example of a game that does not admit (mixed strategy) Nash equilibria. Consider a game in normal form. Let $N=\{1,2\}$ be set of players and $S_i=\mathbb{R}$ a set of possible ...
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1answer
52 views

Facing problem to compute ShapleyValues [closed]

I am facing problem to compute the ShapleyValues. Suppose there are 5 people in a city. First person has 10 dollars, 2nd person has 6 dollars, 3rd person has 11 dollars, 4th person has 12 dollars and ...
1
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1answer
90 views

Expected Utility Method and a Repeated Game Solution [closed]

I am trying to replicate Bruce B. de Mesquita's (BDM) results on political game theory for prediction. Based on where actors stand on issues, their capabilities, salience, BDM's method attempts to ...
2
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2answers
94 views

A game with no pure or mixed strategy equilibrium?

I'm trying to find any and all pure or mixed strategy Nash equilibria for the game $$\begin{array}{|c|c|c|c|}\hline & L & C & R \\ \hline T & (6,2) & (0,6) & (4,4) \\ \hline ...
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1answer
66 views

Dual Core in Cooperative Game Theory

I'm a bit confused over if the dual core of a game is the same as the core of the original game. Definition of dual game: $$ v^*(S) = v(N) - v( N \setminus S ), \forall~ S \subseteq N.\, $$ I then ...
3
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1answer
31 views

Proof that $\min_{b\in B} u(a,b)\leq \min_{b\in B}\max_{a\in A}u(a,b)$

So I have two finite sets $A,B$ and $u:A\times B\rightarrow \mathbb{R}$ a utility function. I am asked to give a certain proof but I don't need help with the whole thing, I just need help figuring ...
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1answer
33 views

Game Theory (continuous utility, pure strategy)

I have a game in which two players, 1 and 2, choose a non-negative real number level of effort $e_1,e_2$ respectively. Their cost for this choice is $ce_i$ for $i=1,2$ where $c>0$ is the same for ...
3
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0answers
90 views

Auction Design : Multiple lots, one win max per bidder, not regret

This is a real life game theory problem. I have to organize an auction. There is a finite number of lots, which are not equivalent. There is a finite number of bidders; the number of bidders is ...
1
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1answer
87 views

Can a transitive relation be represented by a utility function?

I am currently studying for my Game Theory exam and came across a question that seems pretty basic but somehow can't wrap my head around. So if you could share some insight with me, that would be ...
4
votes
1answer
110 views

Have I Found an Error in “Game Theory” by Hans Peters?

I am reading the book Game Theory: A Multileveled Approach Second Edition by Hans Peters. It appears to be the most recent copy. I've search here and on Google for a list of known errors in the book, ...
0
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1answer
58 views

Can a game with a pure strategy Nash equilibrium also have mixed strategy equilibria? [closed]

I have questions: If zero-sum game has pure strategy Nash equilibrium (saddle point), can it have also mixed strategy equilibria? What if game is not zero-sum?
3
votes
3answers
591 views

Playing Odd-Even Cricket, is there a perfect strategy

This is a simple two-player game. One if the people is picked to 'bat'. Both players simultaneous choose a number from 1 to 6. (When playing against a person, you use your hands to show the number). ...
3
votes
1answer
55 views

An inequality relating to moves to P-positions in Nim

I have been researching this variant of Nim. I have been unable to prove the following claim. What is annoying is that I feel I am missing something really obvious. Does anyone have any ideas on how ...
2
votes
1answer
132 views

What is the difference between mixed strategy and behavioral strategy games?

I a beginner in Game theory and reading the book "Non Cooperative Game Theory" by Tamer Basar. I am not able to comprehend the difference between behavioral strategy and mixed strategy. I saw this ...
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1answer
169 views

“Guess the 2/3 of average” game

Let us consider the game "Guess the 2/3 of average" (description: https://en.wikipedia.org/wiki/Guess_2/3_of_the_average) I know, the strategy profile where all players play $0$ is a Nash ...
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votes
2answers
123 views

Is it possible for a strategy game like paper, scissor rock to exist? [closed]

Is there a 2-player strategy game in which 3 (or more) programs don't form a clear dominance structure. That is, program A beats B, B beats C, but C beats A. I'd be more impressed if it was also the ...
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0answers
68 views

Is there a perfect game where there can be no draw and no chance is involved

Is there a game which is perfect, that is: always provides a decisive victor, and involves no component of luck Possible games which would be perfect or near-perfect might involve the pie rule. ...
0
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1answer
43 views

Is 2nd-price with a discount auction truth-telling?

I know that 2nd-price auction is truth-telling, but 3rd-price auction is not. What If I run the regular 2nd-price auction, in the end, the winner is charged at the 2nd bidding price with a discount, ...
3
votes
1answer
128 views

When is the “Taxman Game” winnable?

I recently came across the "Taxman Game" the rules are in the link, but I'll repeat them here: We start with a pile of integers, from 1 to some number that you choose [$n$]. You take one, and I ...
0
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0answers
70 views

Determine the optimal strategies

Player 1 and player 2 is playing a game where player 1 chooses 2 of the following numbers (not the same) 1,2 and 4. Player 2 chooses 1 number also among 1,2 and 4. Neither of the players know what the ...
3
votes
1answer
72 views

Can mixed strategies outperform pure strategies?

Let $G$ denote a game with a finite number $n$ of players in which each player $i$ can choose a mixed strategy $\sigma_i$ over a finite set of pure strategies $\Sigma$. Pure strategies can be seen as ...
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0answers
88 views

Undestranding Basic Game Theory

Lately I'm studying game theory for an exam. I'm having troubles in understanding some theorems since notes I'm studying on are very brief and concise about sense of definition. In this question I'll ...
4
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1answer
113 views

What background is needed to study quantum game theory?

Currently I am learning ( a beginner ) about Bell inequalities and device independent outlook on quantum mechanics. I come across some papers using these concept in quantum game theory. Most of the ...
1
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1answer
53 views

Solving a Recurrence for a Mathematical Game

The problem is: Two players take turns removing coins from a pile. There are initially $n$ coins, and on each turn, a player can remove $a_1, a_2, \dotsc, a_k$ coins. The player who cannot remove ...
3
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1answer
299 views

Optimal strategy for 2 players Lights Out game variation

Consider a turn-based game for 2 players. They're both playing on the same board. The board is 8x8, randomly generated and each cell has 0 or 1 (with equal probabilities), for example: ...
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0answers
31 views

How does the additivity axiom in the characterisations of the Banzhaf and Shapley-Shubik power indices work?

I am struggling to grasp in what sense some power indices - such as the Banzhaf and Shapley-Shubik indices - rely on a linear notion of power. Particularly, why do they satisfy the respective ...
0
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1answer
49 views

Game Theory: Prisoners Dilemma

In $n=2$ person (say $A$ & $B$) prisoner's dilemma, the possible outcomes are $AB, CC,CD,DC,DD$ and the payoffs are $(1,1), (0,3), (3,0), (2,2)$ where $C$ is "cooperation" and $D$ is "defection". ...
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1answer
96 views

Python Integer Game

Jacob and Vicky play the fun game of multiplication by multiplying an integer p by one of the numbers 2 to 9. Jacob always starts with p = 1, does his multiplication, then Vicky multiplies the number, ...
4
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0answers
147 views

Is there a closed-form expression for Shapley value of glove game?

Suppose we have a coalition game with transferable utilities, with $m$ players having a right-handed glove and $n$ players having a left-handed glove. The value of a coalition is equal to the number ...