# 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].

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### Players $A$ and $B$ each pick a number between 1 and 100. The game is a draw if $A=B$, A loses if B-A=1, A wins if B-A>=2. What is the best strategy?

Trying to solve a game theory problem with my son. The game goes like this: Players $A$ and $B$ each simultaneously pick a number between 1 and 100. The game is a draw if both players pick the same ...
1 vote
33 views

### How do you quantify the handicap in this game?

I was playing tennis with my brother. We were playing in a format such that in order to win you have to score $X$ number of points. I gave my brother a handicap. Scenario 1: We play until $7$ points. ...
86 views

### For which set $A$, Alice has a winning strategy?

Alice and Bob are playing a game. They take an integer $n>1$, and partition the set $\{1,2,...n\}$ into two non-empty subsets $A,B$. Alice takes the set $A$ and Bob takes the set $B$. They take a ...
235 views

### What is the math behind the zombie tiktok filter game

I was recently just scrolling my life away on this app called TikTok on this miniature computer called my phone, and recently came accross a trend of people playing a game on a filter. The game is ...
120 views
+50

### Counterfeit money, guaranteeing a profit

Bob and I found two 50 dollar bills out of nowhere. We know they're either both legitimate or both counterfeit. If they're legitimate, they're worth 50 dollars each, otherwise 0. I get one 50 dollar ...
1 vote
62 views

### Expected optimal payoff in a simultaneous 2 player game [closed]

Two players, say $A$ and $B$, play the following game: Both players have $100$ marbles and may put anywhere between $1$ and $100$ marbles in the box each. This decision is not revealed to the other ...
54 views

### Probability of winning in a 20-30 Split Die Game

Alice and Bob have fair 30−sided and 20−sided dice, respectively. The goal for each player is to have the largest value on their die. Alice and Bob both roll their dice. However, Bob has the option to ...
31 views

### Analyzing a Strategic Form Game for Locating a Public Facility

I'm exploring a scenario where a public facility needs to be located along a street segment represented by the interval $[0, 1]$. In this setting, there are $n$ agents, each having their preferred ...
130 views

### Is there a winning strategy in the card removal game?

A and B play a game with blue, red, and green cards. They start with an even number of $n$ blue cards ("stacks"). Player A starts, and the players take turns. Only two moves can be made: (i) ...
26 views

### poker expectation clarification

I ran across a passage in "The Mathematics of Poker" explaining bluffing strategy. Here is a paraphrased version. Two players are playing heads-up ($1$ vs $1$) poker. They're on the final ...
19 views

### Game Theory Question in OR [closed]

Solve the following game between P and Q. Q 6 4 8 0 P 6 8 4 8 8 4 8 0 0 8 0 16 Try with dominance, otherwise you can use any method from game theory.
59 views

### Game: 5 attempts to guess the number between 1 and 100

As the title suggests, I'm playing a game where I have 5 attempts to guess the number between 1 and 100. Each time I guess, the game tells me whether the number is higher, lower, or if I guessed it ...
1 vote
102 views

### Shapley Kernel Proof

I read the paper about SHAP. I think this paper is very interesting ! I would like to understand the algorithm, but I cannot follow the below fact. \begin{equation} X^T WX=\dfrac{1}{M-1}I+cJ \end{...
23 views

### Maximizing the summation of reward of some users (waiting in different positions and lines) using reinforcement learning or other learning methods

There is a mathematical problem that I think can be solved using reinforcement learning and it would be great if you could help me with it. Some users are standing in some lines. There are N lines. In ...
30 views

### Number-guessing game with a pre-known distribution

Let's consider a variation of the famous number-guessing game: a random integer is sampled from $0$ to $15$ with a pre-known distribution $\{P_i\}$. For each guess count, There are two ways to guess ...
21 views

### Optimal control propblem? How to formulated the parameters of following equation?

Recently, I have meet an interesting question? How to let the objective function must be non-negative? Here we have several parameters, for example, $$\dot x(t)=u_1+u_2$$ k_1(x_0,u_1,u_2,T-t_0)=\...
1 vote
94 views

### 20 Sided vs 30 Sided Dice Game [duplicate]

Alice and Bob have fair 30 sided and 20 sided dice, respectively. Alice rolls her die one time. However, Bob can roll his die twice and keep the maximum of the two values. In the event of a tie, Bob ...
17 views

### Relationship between Difficulty of Weakly and Strongly Solving a Game

For a given game, is there any relationship between the computational complexity of weakly solving the game and strongly solving it? I ask because a weakly, but not strongly solved game with simple ...
33 views

### Interesting Solitaire Game [duplicate]

We have three boxes containing finitely many balls (not already empty). The game consists in "emptying" one of the boxes. The only move allowed is to double the content of a box by ...
1 vote
79 views

### Nim variant with two piles where in each turn the sum of their sizes gets reduced by three

So in this game we have two piles of size n and m respectively. Two players take turns, in each turn the player chooses one pile to reduce its size by 2 and the other gets reduced by 1. For example ...
47 views

### Nim variant where the size of the two piles changes [closed]

So in this game we have two piles of size $n$ and $m$ respectively. Two players take turns, in each turn the player chooses one pile to completely remove from the game and then splits the pile that is ...
144 views

### What is the chance of one player winning a chess tournament?

Person A plays a chess tournament against persons B and C. Person B is a professional, while person C is an amateur. Person A wins the tournament if they achieve at least $k$, $k<n$ consecutive ...
63 views

### Is there a closed form for the Josephus Problem on a line?

Consider the following variation of the famous Josephus problem: The numbers 1 through $n$ are lined up. Each round, every $k$th number is eliminated, starting from the first one (so the numbers in ...
1 vote
56 views

### Solving a Certain Combinatorial Game

I've been trying to come up with a combinatorial game even simpler than Hex with non-trivial gameplay and been failing dismally. Currently, my idea is that players sequentially lay pieces on a ...
67 views

### Number of Cards Until 5 Kings Flipped Over Game

"The dealer is known to deal cards from a deck that is comprised of 10 standard card decks. This means that there are 40 of each rank of card and 520 total cards. The dealer plays a game where he ...
1 vote
85 views

### What Hackenbush game represents oof? [closed]

I have been reading about Hackenbush recently and have learned that the surreal numbers can be represented using RGB Hackenbush. I am having a hard time understanding On, Off, and Oof. What Hackenbush ...
39 views

### Weak dominance in second-price auction

Consider a sealed-bid second price auction with two bidders. Standard textbooks claim that bidding one's true valuation $v_i$ weakly dominates bidding lower than one's valuation $b_i<v_i$. But I am ...
77 views

### Market Making Interview Game

I am preparing for an interview with a prop trading firm and wanted to discuss potential strategies for the classic market making games. I have seen similar posts on the forum, but a lot of the ...
18 views

### Finding Nash Equilibrium in Mixed Strategies: Confusion with Dominant Strategies

I'm working with a two-player normal-form game represented by the following matrix: U D L 2,0 2,0 R 6,1 3,2 I attempted to find the Nash equilibrium in mixed strategies and concluded with the ...
1 vote
128 views

### 20 and 30 Sided Dice Game

Two people, A and B, have a 30-sided and 20-sided die, respectively. Each rolls their die, and the person with the highest roll wins. (B also wins in the event of a tie.) The loser pays the winner the ...
30 views

### Polynomial guessing with 2 queries [duplicate]

Alice is allowed to choose an arbitrary polynomial p(x) of any degree with nonnegative integer coefficients. Bob can infer the coefficients of p(x) by only two evaluations as follows. He chooses a ...
274 views

### Is there any winning strategy? [closed]

a) Adam and Eve take turns writing one number each until they have 20 numbers in a row. Adam writes the first number, and the only numbers they can use are 1, 2, 3, 4, and 5. Eve tries to ensure that ...
57 views

### Probability that player B wins [duplicate]

We have two players, A and B, who play a game in several rounds. The game stops when one of them wins two more rounds than the other. What is the probability that B wins if in each round the ...
9 views

### Calculation of NashEquilibria in the selfish routing problem

How should I go for estimating the NE in this problem: I see how to estimate that the optimal is $1/2$ for both trips, however I also see why rational players are incentivized to deviate from such ...
1 vote
28 views

### Iterated Prisoner's Dilemma: Why is All-Defect considered a dominant strategy?

In the finite (e.g. 100 round) Iterated Prisoner's Dilemma, sources like Wikipedia claim that All-Defect is a dominant strategy, which is then proven by backwards induction. I understand this argument,...
138 views

### A game from fault lines in domino tilings

It is known that, for any tiling of a $6\times6$ rectangle with dominoes, there must exist a fault line, or a line cutting the square without cutting any domino. (There is a nice elementary proof of ...
53 views

### Optimal strategy in number guessing game?

Consider the following game. Two players each create a passcode using the digits 0-9 four times with repetition allowed. The two take turns asking a yes or no question about the opponent's passcode. ...
7 views

### Intransitivity of Bounded Dominance For Full-Knowledge Two-Player Alternating Finite Deterministic Games?

Background A full-knowledge, two-player, alternating, finite, deterministic game is modeled by a tree T with labeled leaves and no infinite branches. The game starts at the root of the tree (level 0). ...
25 views

### How many possible games of modified Blackjack?

Assuming we are playing a modified version of Blackjack with the following rules, how many possible games are there? 1 deck of 52 cards (4 suits of A-K) with normal hand total rules (A: 1 or 11, face ...
71 views

### Probability of Losing a Dice Game

Consider the following game. A game involves rolling two six-sided dice, followed by rolling a third six-sided die. To win the game, the number rolled on the third die must lie between the two ...
1 vote
57 views

### 'How much would you pay to play this game' question

'We play a game: I pick a number $n$ from 1 to 100. If you guess correctly, I pay you $n^2$ dollars and zero dollars otherwise. How much would you pay to play this game?' I know the answer for the ...
881 views

### Price of Option in Betting Game

We both put 20 USD into a box. Then, we each generate a number in the interval (0,1) with uniform distribution. The person with the higher number wins and takes 40 USD, whilst the loser is left with 0 ...
54 views

### Best way to win a version of dice black jack.

The game has two players, the player (me) and the dealer. The player starts by rolling a 10-sided die. You roll the die as many times as you want adding all your rolls, your goal is to get as close to ...
1 vote
36 views

### How does $\arg$ relate to the $\operatorname{argmax}$ operator?

I understand the definition of $\max(f(x))$: what $\operatorname{argmax}(f(x))$ means? And I also know that $\arg(z)$ is the angle between the the line passing through the points $z_o\equiv(0,0)$, \$z=...
153 views

### Optimal strategy for chips game

'I have 12 poker chips, we (two people) take it in turns to pick up one or two chips, you win if you don't pick up the last chip. What is the optimal strategy? Do you want to go first or second? What ...
79 views

### Behavior and mixed strategies: extensive form game with perfect recall

In this video there is a game with perfect recall, see the snippet below. They claim that they have an example of a mixed strategy that is not a behavior strategy. However the following holds and is ...
15 views

### Uncertain about solving extensive form of a simple game

I am a complete noob at game theory and trying to complete an exercise which asks to solve the extensive form of a game using backward induction. The game has players 1 and 2 with 2 possible actions ...
1 vote
19 views

### Trying to interpret of Shapley Values

We have been trying out anomaly detection using isolation forest. The model present us with anomaly in our data. We are providing about 25 feature. We want to understand contribution of each feature ...
1 vote