/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q13E Matching pennies. In this simple... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q13E (page 241)

Matching pennies. In this simple two-player game, the players (call them Rand C) each choose an outcome, heads or tails. If both outcomes are equal, Cgives a dollar to R; if the outcomes are different, Rgives a dollar to C.

(a) Represent the payoffs by a2脳2 matrix.

(b) What is the value of this game, and what are the optimal strategies for the two players?

Short Answer

Expert verified

The value of the game is0 and the optimal strategy of both the player will be equal i.e., 12.

Step by step solution

01

Represent the payoffs by a matrix.

(a)It is given that forR to win, the two coins must have same outcome, i.e., either both heads or both tails.

The above condition can be represented as:

H

T

H

+1

-1

T

-1

+1

The matrix represents the moneyR got by game.

H=Head,T=Tail

+1indicates thatR got a dollar whereas-1 shows thatC got a dollar.

02

Calculate the value of this game and the optimal strategies for the two players

(b)

Let, Probability of Rto get head and tail be given asX1and X2.

And, Probability of Cto get head and tail be given as y1and y2.

Max:z: Min:w

z鈮�x1鈭�x2鈥夆赌夆赌夆赌夆赌夆赌夆赌�z鈮�鈭�x1+x2鈥夆赌夆赌夆赌�x1+x2=1鈥夆赌夆赌夆赌夆赌夆赌夆赌夆��x1,x2>0鈥夆赌嬧赌嬧赌夆赌夆赌�

鈥�w鈮�y1鈭�y2w鈮�鈭�y1+y2y1+y2=1y1,y2>0

The value of this game is0 .

Therefore, the optimal strategy of both the player is equal to i.e., 12.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the following network (the numbers are edge capacities).

(a)Find the maximum flow fand a minimum cut.

(b)Draw the residual graphGf (along with its edge capacities). In this residual network, mark the vertices reachable fromS and the vertices from whichT is reachable.

(c)An edge of a network is called a bottleneck edge if increasing its capacity results in an increase in the maximum flow. List all bottleneck edges in the above network.

(d)Give a very simple example (containing at most four nodes) of a network which has no bottleneck edges.

(e)Give an efficient algorithm to identify all bottleneck edges in a network.

Question: A linear program for shortest path. Suppose we want to compute the shortest path from node s to node t in a directed graph with edge lengths le>0.

a) Show that this is equivalent to finding an s - tflow fthat minimizes 鈭�elefesubject to size (f) = 1. There are no capacity constraints.

b) Write the shortest path problem as a linear program.

c) Show that the dual LP can be written as

role="math" localid="1659250472483" maxxs-xtxu-xv鈮�luvforall(u,v)鈭�E

d) An interpretation for the dual is given in the box on page 223. Why isn鈥檛 our dual LP identical to the one on that page?

A quadratic programming problem seeks to maximize a quadratic objective function (with terms like 3x12or5x1x2) subject to a set of linear constraints. Give an example of a quadratic program in two variables x1, x2 such that the feasible region is nonempty and bounded, and yet none of the vertices of this region optimize the (quadratic) objective.

The pizza business in Little Town is split between two rivals, Tony and Joey. They are each investigating strategies to steal business away from the other. Joey is considering either lowering prices or cutting bigger slices. Tony is looking into starting up a line of gourmet pizzas, or offering outdoor seating, or giving free sodas at lunchtime. The effects of these various strategies are summarized in the following payoff matrix (entries are dozens of pizzas, Joey鈥檚 gain and Tony鈥檚 loss).




TONY




Gourmet

Seating

Freesoda

JOEY

Lower price

+2

0

-3


BiggerSlices

_1

-2

+1

For instance, if Joey reduces prices and Tony goes with the gourmet option, then Tony will lose 2 dozen pizzas worth of nosiness to Joey.

What is the value of this game, and what are the optimal strategies for Tony and Joey?

A cargo plane can carry a maximum weight of 100 tons and a maximum volume of 60 cubic meters. There are three materials to be transported, and the cargo company may choose to carry any amount of each, up to the maximum available limits given below.

  • Material 1 has density 2tons/cubicmeters, maximum available amount 40 cubic meters, and revenue \(1,000 per cubic meter.
  • Material 2 has density 1ton/cubicmeters,maximum available amount 30 cubic meters, and revenue \)1,200 per cubic meter.
  • Material 3 has density 3tons/cubicmeters, maximum available amount 20 cubic meters, and revenue $12,000 per cubic meter.

Write a linear program that optimizes revenue within the constraints.
See all solutions

Recommended explanations on Computer Science Textbooks

Fintech

Read Explanation

Computer Organisation and Architecture

Read Explanation

Data Structures

Read Explanation

Issues in Computer Science

Read Explanation

Theory of Computation

Read Explanation

Big Data

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.