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).

learn more… | top users | synonyms

63
votes
0answers
2k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
12
votes
0answers
223 views

The Right Triangle Game

I am looking for a deeper understanding, particularly the optimum strategy and the maximum score as a function of grid size, of the following (single-player) game played with an $n$ by $m$ grid: ($6 ...
7
votes
0answers
142 views

Is there a totally asymmetric 2P $0$-sum game with all payoffs $\pm1$, with a unique Nash eq. which assigns positive probability to each strategy?

Is there a totally asymmetric 2-player zero-sum game with all payoffs $\pm1$, with a unique Nash equilibrium which assigns positive probability to each strategy? By totally asymmetric, I mean that ...
5
votes
0answers
87 views

Apple game question

Player A and Player B play a game. On the middle of the table there is a pot full of $N$ apples of different weights. Player A starts first and chooses an apple and starts eating it. Losing no time ...
5
votes
0answers
237 views

“Infinito”, a combinatorial game with infinite width game-tree

I recently designed a combinatorial game (sequential game of perfect information) with an infinite branching factor, that is it has a game-tree of infinite width. I'm wondering how is it possible to ...
4
votes
0answers
41 views

Who has a winning strategy in the hamilton-circle-game?

The game starts with a graph with $n$ vertices and no edges. The players alternately add edges until the graph contains a hamilton-circle. The player who made the last move loses. Who has a winning ...
4
votes
0answers
66 views

Guess the number of a liar

Sue picks a number from 0 to 3. Tom asks questions about the number, with yes/no answers. For example, "Is it odd" or "Is it 3?" If Sue picked X, she is allowed to lie at most X times. For ...
4
votes
0answers
131 views

algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
4
votes
0answers
78 views

Linear Independence Game

Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
4
votes
0answers
235 views

$n$ players of paper scissor rock

Suppose there are $n$ players $(3\leq{n})$ showing Paper, Scissor or Rock simultaneously. If there is no winner then there is no payoff to any player. If there are winners and losers (e.g. $k$ ...
4
votes
0answers
101 views

Understanding Blackwell's Approachability Theorem

I'm not super solid on my linear algebra, so I am getting lost in the discussions of halfspaces. Can someone give me an intuitive explanation (possibly with a concrete toy problem) of Blackwell's ...
3
votes
0answers
35 views

Simple undetermined games

We know that, under AC, there exists a game in which two players play finite numbers and neither one has winning strategy. There are also such undetermined games when we consider players playing ...
3
votes
0answers
95 views

Maximizing dot-product score by asking queries

Let $a>b>0$, and let $T=\{a,b\}^n$ be the set of all $n$-tuples each entry of which is $a$ or $b$. Let $X\subseteq\{0,1\}^n$ with $|X|>1$, and let $f:T\rightarrow X$ be a function. For each ...
3
votes
0answers
52 views

News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
3
votes
0answers
184 views

Is there any hope to solve the game chess?

I heard about an estimate how many legal positions there are in a chess game. There are roughly ${10}^{40}$. Is it realistic that this amound of positions can be checked in the near future ? Or is ...
3
votes
0answers
42 views

Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
3
votes
0answers
63 views

Terminology questions about a game where one may “save his progress” at the cost of a turn.

The game is for $p$ players who each start at square $1$. Each turn, a player can either roll an $m$-sided dice or place a marker on his current square. If he rolls $x\in\{2,\ldots, m\}$, he ...
3
votes
0answers
106 views

Unexpected hanging paradox maxmin strategies

I have a question about strategies of the players of Unexpected hanging paradox (I am very sorry for a long topic, topic exist already for a while, during this time I try to develop idea how to solve ...
3
votes
0answers
99 views

mixed vs behavior strategies for zero-sum game with infinite extensive form

edit: No responses to this post after a week, so I'm cross-posting it to cstheory.stackexchange here. I'm looking for a known theorem stating that, for appropriate kinds of two-player zero-sum games ...
3
votes
0answers
365 views

Simple game with coins - strategy

Let's play a game: There are $n$ stacks of coins in a row. $i$-th stack consists of $d_i$ coins. Two players: $\text{Player1},\text{Player2}$ make moves alternately. Player in his turn can only take ...
3
votes
0answers
123 views

Optimal strategy in a VCG auction with partial collusion?

Suppose you control the bid prices in a multiple-item VCG auction for a partial coalition of bidders. Each bidder is only allowed to win one item out of the set of multiple items, which are all ...
2
votes
0answers
15 views

dimension of Weber set and selectope (as a operator)

Let $\Omega$ be a finite set of players. For a selector $\alpha:(2^{\Omega}-\{\emptyset\})\rightarrow\Omega$, we define a marginal value operator as a linear operator $m^{\alpha}$ ...
2
votes
0answers
60 views

What is the largest value one can get in game 2048 without new tiles appear

This is a simplified version of the famous game 2048. Given a 4x4 grids with some values chosen from {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048}. A value of 0 indicates that the position in ...
2
votes
0answers
35 views

Who has a winning strategy in “knight” and why?

Perhaps, this game is already known, but I did not find anything about it, I call it "knight". The rules : Player 1 chooses the starting square of a knight on a normal 8x8 - chessboard. The ...
2
votes
0answers
26 views

VCG - plynomial time algorithm when bidders are unit demand

Is there a polynomial time algorithm to run a VCG when bidders are unit-demand? I though to look at the Bipartite graph when the left side is the bidders the right is the items and the edges are the ...
2
votes
0answers
348 views

What is the highest possible score in 2048 hard?

There is a variant of the popular game 2048, called 2048 hard or 2048 impossible, which automatically places each new tile in the hardest possible location. Is this variation possible to solve, and if ...
2
votes
0answers
131 views

What is the optimal strategy for this 2 player game?

Let some finite array of integers is given initially. There is a number k which is initially '0'. In a move, a player will select a number from the array arr[i] and change k to gcd(k,arr[i]). Also, ...
2
votes
0answers
37 views

How to find perfect equilibria in a finite game?

If we define a game with $n$ persons as below: (i) for each player $i$, he has his strategy set $S_i$, $|S_i|=m_i<\infty$, and denote $S=\Pi_iS_i$; (ii) $u_i:S\rightarrow\mathbb{R}$ is a payoff ...
2
votes
0answers
97 views

The name of a game from the 2013 Putnam

Does the following game from the 2013 Putnam have an official name? Based on my searches, it seems to have been created just for the exam. Let $n\geq 1$ be an odd integer. Alice and Bob play the ...
2
votes
0answers
52 views

Optimally Efficient Tournament Strategies

Consider a fair symmetric game between two players that always results in exactly one of the players winning, i.e. there are no ties. When two players $P$ and $Q$ play each other, $P$ wins with a ...
2
votes
0answers
67 views

Is there a game of perfect information that only has a mixed strategy equilibrium?

It is known that certain simultaneous games, say, matching pennies with no pure strategy equilibrium, have a mixed strategy equilibrium. Is simulaneous move at some stage a necessary ingredient of ...
2
votes
0answers
175 views

What is the (expected) outcome of this hybrid auction?

A certain hybrid auction can be accurately modelled as follows. There are $n$ risk-neutral, rational participants $i=1,2,\ldots,n$, and a guy called Zerro: $i=0$. Each, except Zerro, has a private ...
2
votes
0answers
133 views

Comparing Nash equilibrium and Pareto optimal actions

Suppose that $(x_{i}, x_{j})$ identify actions for two players $(i,j)$. If we define Pareto optimal actions by $$h(x_i) +h(x_j)- \eta[p(x_i)+p(x_j)]=2\gamma$$ and Nash equilibrium actions by ...
2
votes
0answers
162 views

A dynamic Stackelberg game - general characterization

my question is about general representation of a dynamic Stackelberg game which is played in continuous time. We have maximization problems of two agents who play this game. Agents are 'Leader' and ...
2
votes
0answers
214 views

Explanation of Mixed Strategy Definition in Game Theory

Definition: Let $(N, A, u)$ be normal-form game, and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$. Then the set of fixed strategies for player $S_i=\Pi(A_i)$. ...
2
votes
0answers
138 views

Game Theory - Extensive Zero-Sum Game Property Proof

How I might I go about to prove (or disprove, but I believe that this is true) the following: We call a 2-player extensive game $\Gamma$ a zero-sum game if the sum of the 2 payoffs for an terminal ...
2
votes
0answers
108 views

proof using (fixed point theorem)

I am seeking to solve for a Nash equilibrium in pure strategies $(d_2,d_2)$ involving two players, $1$ and $2$. Given that $h'(.)$ is s strictly decreasing and continuous function, $\Phi(d_1-d_2)$ ...
2
votes
0answers
146 views

Is Bingo Games Solved?

So we have numbers 5 rows by 5 columns. Different players have the same table. The numbers on each cell of the tables are different. Different players choose which numbers will be marked. We select a ...
2
votes
0answers
679 views

How can I bid optimally on bitcoinauction.us?

The website www.bitcoinauction.us hosts a bidding game with a prize. Note that it's not a real auction because you don't get your bid back if you're outbid. The game works as follows: The game ...
1
vote
0answers
16 views

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 ...
1
vote
0answers
10 views

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 ...
1
vote
0answers
30 views
+50

Aumann-Shapley Uniformly Better Principle

Let $n_1,..,n_r$ be $r$ positive integers, and let $1 \leq k \leq n$, where $n=n_1+...+n_r$. Consider an urn containing $r$ different types of balls, $n_1$ balls of type 1, $n_2$ balls of type ...
1
vote
0answers
27 views

quasi rationality, interesting axiom of revealed preferences

So imagine there is a notion of rationality that captures the idea of "thresholds in preference." For example, let $\mathbb{Z}$ be the integers: $\mathbb{Z} = \{\dots, -10, -9, \dots, 0, 1, 2, ...
1
vote
0answers
57 views

Game theory: connect four?

Through Allis' solution etc... P1 can force a win if he places the first stone in the highlighted region. Assuming both players have perfect information, it will take 41 turns maximum (if I recall ...
1
vote
0answers
88 views
+50

Game theoretical approach to other branches of mathematics

Are there some methods and ideas derived from game theory that are successfully applied to better (or more intuitively) understand theorems and proofs or tackling problems from other areas of ...
1
vote
0answers
34 views

A conjecture about Nash Equilibria in multiplayer games involving card drafting

A deck of $N$ cards is used to play a 4-player game. The game begins with each player being randomly dealt 7 cards from the deck. They then take turns according to a set of rules, after which a single ...
1
vote
0answers
50 views

Putnam game theory question

There are n>=1 boxes in a line were n is an odd integer. Two players, Connor and Andrew, are playing a game. On 'your' turn, place a stone in a box OR take a stone out of a box and place a stone in ...
1
vote
0answers
16 views

Prove that a mixed strategy in two player, zero sum, matrix game must exist (alternative proof)

So I am having a trouble with this game theory proof. I feel pretty good with my answer for part 1, but I am not really sure how to get started on the rest of it. Any help would be appreciated. Let ...
1
vote
0answers
43 views

Uniqueness of solution for a system of differential equations

A friend of mine working on Auction Theory needs to establish uniqueness of solution (up to initial and boundary conditions) of a system of differential equations of the form $$ ...
1
vote
0answers
14 views

Distribution of coalition cost among coalition members (game theory) on the basis of contribution in coalition

Does any one know or point out the method or technique used for the distribution of the coalition cost among the coalition members depending upon their contributions in the coalition. In other words, ...