0
votes
5answers
80 views

Some examples of applications of Game Theory

I'm approaching my junior year of HS now, and I'm looking for a good science fair project to do. I love mathematics, so I decided to a category of mathematics that can help base logical conclusions to ...
0
votes
1answer
29 views

Board Game Markov Process - Transient Probabilities

I need to write an essay on the Game of Life board game, and so I studied up on Markov Chains to help me calculate the probabilities and average payoffs for the spaces; however I'm not sure whether ...
2
votes
1answer
35 views

Game theory: Mixed Strategies and Nash Equilibrium

So I've recently become interested in game theory, and I've visited this site to help me understand what exactly game theory is and the applications of it. In the lesson, they use an example of ...
1
vote
2answers
36 views

Finding optimal mixed strategy

I have to find an optimal mixed strategy for the 'column' player, who mixes with the probabilites $q_1,q_2,q_3$. What is known is the optimal mixing of the 'row" player. The game is a zero-sum game, ...
1
vote
1answer
96 views

Game Theory - Nash equilibrium question

Consider a game in which 2 players transmit packets in a network with a selected power $x ∈ [1, A]$ and $y∈ [1, A]$, respectively. The utility of the players can be expressed as: $$u_{i} (x,y) = ...
0
votes
0answers
108 views

Robinson Crusoe economy problem

Consider the Robinson-Crusoe one-consumer, one-producer economy. Compute the equilibrium prices, profits and consumption when the production function is $f(L)=\sqrt{L}$, the utility function is ...
0
votes
0answers
80 views

Game Theory Question - Matching Coins

I'm a third year maths undergrad just started a topic on Game Theory. I have just approached a question, and after drawing my tree diagram I have to indicate information sets. The question is as ...
1
vote
1answer
56 views

Coin based subtraction game

I'm having a problem in Game Theory where I am trying to understand how a subtraction game can be interpreted by a coin based game. From my book: The problem I'm having is if I have 9 coins and the ...
1
vote
1answer
50 views

Questions about Auctions

I am having a hard time figuring out a problem. In a first price auction with a reserve price R and values of the bidders are U[0,1], how do we find expected revenue given the strategy of both of them ...
0
votes
1answer
50 views

How to maximize your score in this game?

Consider the following game: On a table, there are two glass bowls with $x$ and $y$ beads each. In each move you can do the following: Remove a bead from the first bowl and put it in the second ...
1
vote
2answers
81 views

Second-Price Sealed-Bid Auction

Consider 2 individuals who are interested in one indivisible object. Each player $i$ has a valuation $v_i > 0$ for the object. Assume $v_1 \geq v_2$. In this scenario, each player submits a bid ...
1
vote
0answers
124 views

Subgame Perfect Nash Equilibrium Problem

Suppose that there are two incumbent icecream vendors, but that there is a possibility of entry of a third vendor. Specifically, at any location on the beach a third vendor can, after observing the ...
3
votes
1answer
40 views

Centipede game of 2 students

PROBLEM We have two students who both begin with 1 point, their utility is equal to the number of points they have. Student $1$ has the first turn, he can decide to end the game and both students ...
0
votes
1answer
371 views

Ice cream vendor problem

PROBLEM This is a question considering game theory. Assume there is a beach with $n$ ice cream vendors on it who position themselves along the beach. For an arbitrary $n$, find a candidate Nash ...
2
votes
1answer
84 views

Coin game strategy [closed]

PROBLEM Suppose two players play a coin game. They both have a coin and can choose themselves what the probability is that they will play head $H$. So for player $i$, the chance he gets head is ...
3
votes
1answer
53 views

Static game question

GAME PROBLEM At a tv game show, two players are handed six cards each. Each of the six cards shows a number, and both players have the same cards. The six numbers are $0,0,3,0,15$. The game is as ...
2
votes
1answer
79 views

Finitely repeated games

Consider the following matrix game: \begin{matrix} & L &M&R \\ T&8,8 & 0,9 & 0,0 \\ C&9,0 & 0,0 & 3,1 \\ B&0,0 & 1,3 & 3,3 \end{matrix} For ...
0
votes
1answer
207 views

Question related to the General equilibrium in exchange economies

I need some help in solving this question: Consider the following two-person, two-good economy. Persons $A$ and $B$ each consume two goods xylophones $(x)$ and yams $(y)$. Person $A$ enters the world ...
0
votes
1answer
88 views

Consider a game in which two players take turns removing any positive number of pebbles they want from one of two piles of pebbles.

Consider a game in which two players take turns removing any positive number of pebbles they want from one of two piles of pebbles. The player who removes the last pebble wins the game. Show that if ...
2
votes
1answer
158 views

Duality and the Minimax Theorem

I review LP duality by reading Lecture 7: The LP Duality Theorem. I get the idea how to find the dual LP from primal LP, but my basic knowledge is not enough for finding dual LP for the LP in chapter ...
0
votes
0answers
118 views

Routing Game : Modified Pigou's example

Suppose that we modify Pigou’s example so that the lower edges have cost function $c_1(x)=1$ and $c_2(x) = (x/n)^d$ for some $d ≥ 1$. What is the price of anarchy of the resulting selfish routing ...
0
votes
1answer
180 views

Air Strike Game

This is an Air Strike Game with the solution, I have added some questions regarding the solution and I would appreciate if someone could answer them. Army $A$ has a single plane with which it can ...
4
votes
1answer
62 views

Generlized Büchi Games and Closed under superset Muller Games

For a unique infinite play $p$ in a 2-Player game $G=(V_0,V_1,E)$. Let $$ \inf(p) \subseteq V_0 \cup V_1 $$ be the set of vertices which occur infinitly often in $p$. Generlized Büchi (GB) Games ...
3
votes
2answers
222 views

Exercise in Mechanism Design

I found an exercise with solution in the field of Mechanism Design. The problem is I don't understand the solution. Exercise. Use the characterization of incentive compatible direct-revelation ...
3
votes
1answer
120 views

Existence of asymmetric equilibria in the dollar auction game

Consider a game in which an auctioneer sells one dollar to the highest bidder. The high bidder wins the dollar, but every bidder pays their bid. Concretely, assume that there are two bidders ...
0
votes
1answer
66 views

Finding a dual Linear-Program

We are trying to prove Von-Neumann's MINIMAX Theorem namely $$\max_{x\in\Delta_{n}}\min_{y\in\Delta_{m}}y^{T}Ax=\max_{x\in\Delta_{n}}\min_{1\leqslant i\leqslant n}(Ax)_{i}$$ (Here $\Delta_k$ is the ...
0
votes
1answer
128 views

Find profit maximizing profit and quantity given willingness to supply curves and a merger

This is a homework problem, but I'm at my wit's end. I don't even know where to start on this, but I've tried a number of strategies. Consider a regional market for wholesale electricity where ...
2
votes
2answers
2k views

Cournot-Nash Equilibrium in Duopoly

This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way of solving the following question: There are 2 firms in an industry, ...
-4
votes
1answer
403 views

Nash equilibria and best response functions

a) Let $G=(A,u)$ be a strategic game such that, for each $i \in N$ $A_i$ is a nonempty, convex, compact subset of $R^{m_i}$ $u_i$ is continuous For each $a_{-i}$, $u_i(a_{-i}, . )$ is quasi-concave ...
0
votes
1answer
716 views

The core/Shapley value

Please help me to calculate the core of this easy coalitional game. I really didn't get it from my game theory course but want to understand the mechanism of calculating, describe it in detail please! ...
2
votes
0answers
130 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 ...
0
votes
0answers
86 views

Can the Nash bargaining solution be applied in repeated game?

I am trying to develop a model involving two agents who interact strategically to set an optimal time for a joint work. These agents will have to meet repeatedly. I want to derive the optimal time for ...
2
votes
1answer
144 views

vickery auction question(second-price auction)

The question is as follow, Alice and Bob would both like to own the same manuscript. The manuscript is worth 5 million to Alice and worth 3 million to Bob. The present owner of the manuscript ...
0
votes
1answer
227 views

subgame perfect nash equilibrium for war of attrition

the question is as follow: suppose that two players are playing war of attrition, that means both of them could choose either to fight or quit, if either one of them quit, the game ends, and if ...
1
vote
2answers
3k views

cournot competition with N-firms

The question is as follow: Here is how we can think of N-firm Cournot competition. Assume all the firms have the same marginal cost C > 0. Firm 1 chooses Q1, Firm 2 chooses Q2, and so on. The market ...
0
votes
1answer
2k views

cournot equilibrium and stackelberg equilibrium question

Question is as follow: there are 2 firms that want to enter the apple juice market in country A. There are no existing firms in the market or potential entrants. They need to decide on yearly ...
1
vote
1answer
152 views

Cournot-game problem

I'm so stuck with an exercise about Cournot game and was hoping if someone could help me out here. Would appreciate all the help. This is the exercise: Consider the market for Blue Turtle (a new ...
2
votes
1answer
180 views

mixed strategy nash equilibrium question!

Suppose the game consists of only $2$ players, player $1$ and player $2$, and each of them has only $2$ strategies to choose between. This gives a $2$ by $2$ payoff matrix. Player $2$ has no ...
1
vote
1answer
211 views

Question on mixed nash equilibrium!

The question is as follows: Think of the Golden Ball game. Now player 1 is money-minded and jealous, and player 2 is very good-hearted, so the payoff matrix is follows: ...
2
votes
0answers
132 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 ...
0
votes
1answer
301 views

Relevant Topics for Presentations

I have to make an presentation as a part of our math course. We are provided relavent topics but I am looking for somethat that is challenging, related to economics something like game theory or ...
1
vote
0answers
225 views

Carrier of a mixed strategy in a 3x3 bimatrix game

We are given a bimatrix game (A,B) = $\begin{bmatrix}0,4 & 4,0 & 5,3\\4,0 & 0,4 & 5,3\\ 3,5 & 3,5 & 6,6 \end{bmatrix}$. Suppose (p, q) is a Nash equilibrium in (A,B). Prove ...
0
votes
2answers
621 views

Subgame Perfect Nash Equilibrium

My homework question is summarized below: There are 7 players (say P1,P2,...,P7) trying to split 100 dollars. The game starts with P1 proposing an allocation of the 100 dollars to each ...
0
votes
1answer
701 views

Finding Nash Equilibria with Calculus

The problem is summarized as: There are two players. Player 1's strategy is h. Player 2's strategy is w. Both of their ...
1
vote
1answer
846 views

Unable to find Nash equilibria in mixed strategies

Here is the strategic form game: Player 2 Left Middle Right Top 2,2 0,0 1,3 Player 1 Middle 1,3 3,0 1,0 ...
6
votes
2answers
317 views

(Game Theory) Incomplete Information extension of the 'Centipede' Game

This question is an extension of the Centipede Game. My prof. posed this to me in class and I can't figure out how to approach this problem. Imagine in this game, there is an alternative ...
1
vote
1answer
154 views

Game Theory- Second-Price Sealed Auction- 2 Players

Assume Player 1 has valuation 400. Player 2 has valuation on the interval [0, 100]. What would the equlibrium be? Assume Player 2 knows Player 1's valuation, but Player 1 knows only the probability ...
1
vote
1answer
168 views

Nash equilibria of mixed strategies

I am given the following game to find nash equilibria in pure and mixed strategies: $\begin{pmatrix}& & Litte John &\\ & & c & w \\Big John & c & (5,3) & (4,4) \\ ...
4
votes
2answers
2k views

Cournot Nash Equilibrium Between Two Firms

Suppose we have two firms with specialized, but similar products. Suppose market demand for the two products is: $$p_1(q_1,q_2)=a-bq_1-dq_2$$ $$p_2(q_1,q_2)=a-bq_2-dq_1$$ where $d \in (-b,b)$. Suppose ...
1
vote
1answer
138 views

A strategy in a possibly-infinite game determined by coin toss

A pair of positive integers $x_0$ and $y_0$ are selected and further pairs are generated according to the following rule: After the values of $x_n$ and $y_n$ have been found, a coin is tossed. If it ...