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Questions tagged [game-theory]

Theory of dynamic processes with several competing actors that try to achieve some goal in a strategic way.

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Is there any model of Game of Life compatible with hypercomputation?

I found a question in Mathematics Stack Exchange which asks a very similar question (https://math.stackexchange.com/questions/1023812/hypercomputation-higher-dimensional-variants-of-conways-game-of-...
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1answer
34 views

Bertrand's ballot theorem

I want to understand the dynamic programming equation of https://en.wikipedia.org/wiki/Bertrand%27s_ballot_theorem theorem. it is this If i number of people voted for A and j number of people voted ...
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34 views

Computing Nash equilibria in discrete auctions

I am trying to compute the (pure strategy) Nash equilibria of some discrete auctions. More precisely, let us define the strategy of each player as a function mapping from every valuation that they ...
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Game Theory: Using a convex hull algorithm to map out Pareto outcomes

I have started studying the Pareto efficiency notion in Game theory. The definition I am familiar with is this: Strategy profile $\mathbf{s}$ Pareto dominates strategy $\mathbf{s}'$ if for all $i\in\...
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Must the champion of an entire tournament beat the champion of a possible tournament among other players?

I have a list of "players" of a "tournament". Any two adjacent players may "compete", which results in the loser being thrown out of the tournament. Winning is not transitive. The winner of a given ...
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46 views

Can Pareto Optimality be compared to Nash Equilibrium?

Given a state $s$, and a value function $v^i$ that determines the expected payoff for the i-th agent in that state, can the two definitions below, one of Nash equilibrium and another of Pareto ...
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26 views

Algorithm for first-price, sealed bid simultaneous auctions for distinct items and budget-constrained bidders

Context: I play in a simulated online basketball league (a la the NBA) where each human player controls one of the teams. When each simulated basketball player is a free agent (that is, their previous ...
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30 views

Transformation of a Product Buchi Game to a Parity Game

Is there anyway to express a Product Buchi game as a parity game? There is no stochasticity in my original turn-based game and a Deterministic Buchi Automaton is constructed for LTL specifications.
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How to express the existence of winning strategy of the starter of a game in temporal logic?

Consider a two-player game. A winning strategy of a player is a strategy following which the player can always beat his opponent, no matter how his opponent responds. A game can be unfolded to a ...
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2answers
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Determining the existence of a forced win vs determining the best outcome

A large body of work looks at the computational complexity of games. In particular, identifying a forced win from a given position. Is this problem equivalent under polynomial-time reduction to ...
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3answers
157 views

How does “solving” a game like connect four or tic-tac-toe work?

To mathematically solve a game you have to prove, using various tehniques, that some player will win, lose or draw the game. Specifically I'm interested in solving games by brute force (trying out all ...
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1answer
77 views

Optimal strategy in repeated cake-cutting

Alice and Bob play the famous game of divide and choose $n$ times with $n$ identical cakes. Each time, Alice cuts the cake to two pieces, and Bob chooses the piece he prefers. Alice does not know Bob'...
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42 views

Agreeing on a Time in a Trustless Network

Imagine a chord-like network $N$. each machine $m \in N$ is directly connected to $N_m \subset N$. Without relying on official timeservers, I want all machines to synchronize to some time with ...
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1answer
120 views

Variant of nim game with particular rules

Let's say we have the following rule for that particular Nim game: You have multiple heaps, and you have to remove 1,2 or 3 sticks (not more) from a SINGLE choosen heap. The player who pick up the ...
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177 views

What is the best heuristic function for evaluating the state in game of sprouts?

We are developing game of sprouts bot using min-max algorithm. We need a good heuristic function for evaluating the state of the game/player. Can someone please suggest?
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201 views

Utilization of static evaluation functions in chess

Disclaimer: I'm very new to computer science & game programming I'm having trouble with understanding how to implement a static evaluation function in chess. The logic seems simple like just ...
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1answer
564 views

Unique Stable Solution in Stable Marriage Problem: Is it Pareto-efficient and a Nash Equilibrium?

The deferred acceptance algorithm solves the Stable Marriage Problem in a two-sided network, where each agent has complete preferences over each agent of the other side. There is always at least one ...
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85 views

Analysis of the Banana Game

My computer science professor introduced an interesting game in order to get us (his students) more familiar with the Stack and Queue ADTs. Game Description The banana game is played with a ...
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1answer
38 views

Probability that a leaf with 1 will be selected in game tree evaluation

I am trying to understand randomized AND-OR Game Tree Evaluation. I am stuck with proving the most basic case, namely, an OR node with two leaves (AND node with two leaves is similar). ...
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315 views

Board Game algorithms

I have to find an evaluation function in which we evaluate the state of the game in each round. The game is like " roborally " , it is 2 player-Game in which there are a board and cards ...
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54 views

Simultaneously optimal strategies for both players in zero-sum games

Given a matrix $A \in \mathbb{R}^{m \times n}$ that describes a zero-sum game. What are necessary and sufficient conditions for cases in which the k-th pure strategy of the row-player and the l-th ...
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Are dynamic and agent-based network formation models exclusive groups?

I am studying network formation models and I have seen the separation of those models into two groups (wikipedia): Agent-based models: Aka game-theory based models. This distinction makes sense to ...
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how to use the solution of mean payoff game to arrive at winning regions of parity game?

The 7th chapter (7.3 section) of "Automata Logics and Infinite Games" by Erich Gradel, Wolfgang Thomas, Thomas Wilke gives a way to convert max parity game to mean payoff game. I do not understand ...
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158 views

Game of Nim (Find winning states in large range)

If I want to solve a game of nim with 3 piles containing (x, 2x, 3x) respectively I know that the solution of (winning) iff x^2x^3x != 0 but if x can be in the range from 1 to 2^61 so How many ...
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51 views

Develop a winning strategy for a game

Okay so I've been going crazy about this problem I was given. It is as follows: Given a pile of N matches, 2 people take turns removing matches from the pile. When it is your turn you are allowed ...
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Is this modified version of Nim solvable?

In the regular version of the game Nim, you have $n$ piles of stones, and you may remove one or more stones from a single pile in your turn. A modification I thought of involves the concept of piles ...
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1answer
154 views

Analysis of komi values for increasing Go board sizes and agents strength

Komi is the additional number of points given to the non-starting player in the game of Go. For 19x19 board, currently it is 6.5 under Japanese rules, 7.5 points under Chinese rules. In the past it ...
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Degrees of Separation Minimization & Pathfinding Elimination Strategy for Large Networks

Assume you have a graph/network N; Boundary Conditions: each vertex/node $n \in N$ can support a maximum number of $e_{max}$ connections Nodes can join and leave the graph. Each node $n$ has an ...
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1answer
313 views

Machine learning approach to auction game

I am newbie with machine learning. In order to learn more I decided to try solving a specific problem/game that I have in mind. The problem is the following: I have a list of $N$ items which are ...
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1answer
69 views

Algorithm to find the optimal strategy to feed your cousin (Long question)

I have a programming problem, instead of describing the whole domain it's easier for me to create an analogy. The numbers I use in this example are completely arbitrary. Your cousin is coming over ...
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1answer
113 views

Understanding Incentive Compatibility of pooled Bitcoin Mining paper

I'm trying to understand the paper Incentive Compatibility of Bitcoin Mining Pool Reward Functions (Schrijvers, Bonneau, Doneh and Roughgarden, in Financial Cryptography and Data Security – ...
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1answer
316 views

How are the 1s and 0s derived by the minimax algorithm for the game called “nim”?

I'm having trouble figuring out why the 1s and 0s are where they are in this game tree for the game Nim: There is a statement in the book that explains the numbers but it's still not clear to me, In ...
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118 views

Algorithm for calculating the Grundy value of a green-hackenbush graph

I need an algorithm to convert a green hackenbush graph to it's equivalent tree so that I can find the Grundy value of the Graph using colon principle. I thought of finding the bridges in the given ...
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An impartial game of snake

There is a display of NxN (N<=90) pixels with some blocked pixels and a snake of width = 1 pixel and variable length and two players A and B play a game. The game proceeds as follows - Both ...
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1answer
138 views

Solving the 10 players duel game

Players #1 to #10 with hit probabilities $0.1,0.2,...,0.9,1$ (which are common knowledge) take part in a shooting game. Rules of the game: Players from #1 to #10 take turns to fire at each other. ...
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Is a rational game playing agent still rational when not staying off terminal states when it looses in all child states?

I am working on a programming assignment to implement a Minimax algorithm and heuristic for the game Lines of Action. The rules are pretty simple, each player alternates turns moving a single piece. ...
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670 views

Representation of 8-Puzzle for (A*) Search Algorithm in C

I am trying to complete CS50's hacker edition pset3, where you have to make a program that will solve the game of eight or fifteen from its current state, this must be written in C. https://en....
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Terminology for non-game theoretic techniques

When solving an adversarial problem there are two basic approaches: one that takes into account the thinking process of both sides, and as opposed to that the non-psychological computations. For ...
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357 views

A nim game variant with pass move

This variant is almost similar to the normal nim game which states - Two players take turns to remove one or more items from a single, non-empty pile. The player who removes the last item from the ...
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1answer
136 views

Nim Games: Is it possible to reduce following using Grundy Number?

Following is the problem from a finished programming contest. Two players try to create a number, DesiredTotal, by adding numbers from 1 to N. The first to cross the number is the winner. The ...
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134 views

Complexity of parity game solving compared to PLS, PPA, and PPAD

Since parity game solving is in TFNP ("Total Function Nondeterministic Polynomial") (and the decision version is is NP ∩ coNP), I wonder whether it is contained in PLS ("Polynomial Local Search") or ...
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30 views

Parity games and acceptance conditions of automata on infinite words

I would like to understand the ideas behind parity games and their winning conditions. Where do they come from?
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1answer
465 views

Evaluation functions of Minimax algorithm

Let's say we have the following relationship between $f_1$ and $f_2$: $$f_2(s) = \sqrt{1 + f_1(s)}$$ And $f_1$ returns a positive value. Why is it that minimax search using $f_2$ is guaranteed to ...
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Number of winning combination in Nim

I am studying game theory and I am wondering how many winning combinations are possible for Nim game? Suppose, stones = 500 and piles = 5. With these number, there are many initial game positions are ...
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76 views

Computing losing positions in modified Wythoff's game efficiently

Wythoff's game is as follows: there are two players $A$ and $B$ ( $A$ being the first player ) and there are $2$ piles of stones. When his turn a player can remove one or more stones from anyone pile ...
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1answer
276 views

Reccurrence for the game of pile of stones

I am trying to solve this question from Project Euler for past few days: Divisor game. The problem is as follows: Two players are playing a game. There are $k$ piles of stones. When it is his turn ...
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1answer
52 views

In Minimax, how should we handle draws?

A player playing the Minimax strategy should choose moves which minimise their maximum loss. What should happen when draws can happen? Should we class the draw as a win because we aren't losing? Or ...
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629 views

Winning strategy of Nim game when picking from multiple piles is allowed

I am studying with Game theory right now. In Nim game, in any turn, a player can move any number of stones from any one pile. I am wondering what might be the winning strategy of first player if in ...
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102 views

How are benchmarks created?

Computing equilibria in games and the complexity thereof is imho still quite a young field in which a lot of work still is to be done (especially the former). GAMUT (2004) is a very nice "suite of ...
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121 views

Metaheuristic for NP-complete problem without exact algorithms other than brute-force

Computing Pure Nash Equilibria (PNE) is a Game Theory related problem. Deciding if there exists PNE in a given game has been shown to be NP-Complete (Gottlob et al.). I want to design a metaheuristic ...