A and B play the following game: A writes down either number 1 or number 2,...

A and B play the following game: A writes down either number 1 or number 2, and B must guess which one. If the number that A has written down is i and B has guessed correctly, B receives i units from A. If B makes a wrong guess, B pays i unit to A. If B randomizes his decision by guessing 1 with probability p and 2 with probability 1 - p, determine his expected gain if

(a) A has written down number 1 and

(b) A has written down number 2.

What value of p maximizes the minimum possible value of B's expected gain, and what is this maximin value? (Note that B's expected gain depends not only on p, but also on what A does.)

Consider now player A. Suppose that she also randomizes her decision, writing down number 1 with probability q. What is A's expected loss if

( c) B chooses number 1 and

( d) B chooses number 2?

What value of q minimizes A's maximum expected loss? Show that the minimum of A's maximum expected loss is equal to the maximum of B's minimum expected gain. This result, known as the minimax theorem, was first established in generality by the mathematician John von Neumann and is the fundamental result in the mathematical discipline known as the theory of games. The common value is called the value of the game to player B.

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Write a program with at least 2 functions that play the game of “guess the number”...

Write a program with at least 2 functions that play the game of “guess the number” as follows: Your program chooses the number to be guessed by selecting an integer at random in the range 1 to 1000. The program then displays the following: I have a number between 1 and 1000. Can you guess my number? Please type in your first guess. The player then types the first guess. The program then responds with one of the following: 1. ...

Two persons (A & B) play guess the number game. First person choses an integer in...

Two persons (A & B) play guess the number game. First person choses an integer in the range [1,10] and second person has to guess it by asking questions that the first person can only answer in yes or no. a) Person B asks questions in the sequence, “is it x?” where x=1, 2, …, 10 e.g. first question would be, is it number 1. If no, 2nd question would be, is it number 2, etc. Find the expected number...

Consider the following two-player game. X Y A 1, 2 3, 2 B 2, 0 1,...

Consider the following two-player game. X Y A 1, 2 3, 2 B 2, 0 1, 1 a) Find the best response function for player 1. b) Find the best response function for player 2. c) Graph the best response function for both players, putting either p1(A) or p1(B) on the x-axis and putting either p2(X) or p2(Y) on the y-axis. d) Label all Nash equilibria on the graph from part c. e) Clearly state all Nash equilibria.

This game will play like this : 1) You start with $100 2) You roll a...

This game will play like this : 1) You start with $100 2) You roll a regular six-sided die: { 1 or 2 --> I multiply your money by 1/5 } { 3 or 4 --> I take all your money} { 5 or 6 --> I multiply your money by 6 } 3) With whatever money you have left, we repeat step (2) one more time 4) Then we stop Find the expected amount of money you will have at...

Two players A and B play a game of dice . They roll a pair of...

Two players A and B play a game of dice . They roll a pair of dice alternately . The player who rolls 7 first wins . If A starts then find the probability of B winning the game ?

In the game where A, B and C play together with dice. A rolls first., then...

In the game where A, B and C play together with dice. A rolls first., then B, then C and again A, B,... What is the probability that A is the first person that flips 6 first? A. 0.4727 B. 0.2379 C. 0.3956 D. 0.5

A game of chance offers the following odds and payoffs. Each play of the game costs...

A game of chance offers the following odds and payoffs. Each play of the game costs $100, so the net profit per play is the payoff less $100. Probability Payoff Net Profit 0.10 $700 $600 0.50 100 0 0.40 0 –100 a-1. What is the expected cash payoff? (Round your answer to the nearest whole dollar amount.) a-2. What is the expected rate of return? (Enter your answer as a percent rounded to the nearest whole number.) b-1. What is...

1. Feedback is an essential component of game interfaces. Play an electronic game for at least...

1. Feedback is an essential component of game interfaces. Play an electronic game for at least 1 hour. How does this game provide feedback to the player through interface design? Is there any missing information that you feel should have been conveyed? What would need to be modified in order to convey this information? 2. How do aesthetics play a part in providing feedback to the player? Analyze the aesthetics of a game’s interface. Do the aesthetic components enhance or...

You pay $1 to play a game. The game consists of rolling a pair of dice....

You pay $1 to play a game. The game consists of rolling a pair of dice. If you observe a sum of 7 or 11 you receive $4. If not, you receive nothing. Compute the expected value and standard deviation for this game?

Using the models and theories of either (1) market structure, conduct and performance, (2) game theory...

Using the models and theories of either (1) market structure, conduct and performance, (2) game theory or (3) the boundary of the firm, perform a microeconomic analysis of one appropriate economic issue or phenomenon of your choice. Clearly explain your chosen question, method and conclusions. Comment on the link between market structure and performance for your chosen issue. - The issue is the effect on the Australian Meat market as a result of COVID - 19 - The word limit...

Subjects