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math, economics, game theory

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Two people have 10 dollars to divide between themselves. they use the following procedure. each person names a number of dollars (nonnegative integer), at most equal to 10. IF the sum of the amounts that the people name exceeds 10 and the amounts named are different, then the person who named the smaller amount receives that amount and the other person receives the remaining money. if the sum of the amounts that the people name exceeds 10 and the amounts are the same, each person receives 5 dollars. determine the best response of each player to each of the other players' actions and thus find the nash equilibria

I need to know how this answer is gotten, so plz show or tell me how work done

players : two individuals
actions : each players' set of actions is the set of effort levels (non negative numbers)
preferences : player i's pereferences are representede by the payoff function Ai(c+Aj-Ai)

To find the nash equilibria we can construct and analyze the players best response functions. given Aj, in dividual i's payoff is a quadratic function of Ai that is zero when Ai=0 and when Ai=C+Aj, and reaches a maximum in between. the symmetry of quadratic functions implies that the best response of each individual i to
Aj is Bi(Aj) = 1/2(C+Aj)

if u know calc, you can reach the same conclusion by setting the derivative of player i's payoff with respect to Ai equal to zero.

  • math, economics, game theory -

    Two people have 10 dollars to divide between themselves. they use the following procedure. each person names a number of dollars(nonnegative integer) at most equal to 10. if the sum of the amounts that the people name is at most 10, then each person receives the amount of money she named and remainder is destroyed. if the sum of the amounts that the people name exceeds 10 and the amounts named are different, then the person who named the smaller amount receives that amount and the other person receives the reemaining money. if the sum of the amounts that the people name exceeds 10 and the am ounts named are the same, then each person receives 5 dollars. determine the best response of each player to each of the other players' actions to find the nash equilibria.

    a guideline on how the book says to do this is posted in the previous question i did about this...just scroll down a bit to MATH-eric

  • math, economics, game theory -

    First of all, Nash Equilibrium analyses are pretty advanced stuff. I understand the basics. However, I could be out of my league with anything advanced. Especially if you want to apply some differential function (calculas) to possible responses as you suggest in your earlier post.

    As you probably have discovered, the hard part to a nash equilibria is deciding what the other guy's initial position will be. Once that's done, everything falls into place.
    I need to think about this problem some more.

    Sorry for the lack of specifics, and lottsa luck.

  • math, economics, game theory -

    hmm...thanks, those are along the same lines as my thoughts, which is not helpful for turning this problem into my prof @_@

  • math, economics, game theory -

    1) If C = 1000 + 7/8[GDP-1000], I = 700 and G = 1000 and the economy is currently in equilibrium at 400 below full employment GDP, the correct fiscal policy would be to increase G by?

    2) If C = 500 + 3/4[GDP- 100], I = 300, G = 400, Xn =- 10 and full employment GDP is 210 less than current GDP, the proper action would be to increase taxes by?

  • math, economics, game theory -

    400

  • math, economics, game theory -

    If C = 500 + 3/4[GDP- 100], I = 300, G = 400, Xn =- 10 and full employment GDP is 210 less than current GDP, the proper action would be to increase taxes by?

  • Economics -

    1) GDP=14600
    2) GDP=1360 ,but I do not know what it means "the proper action would be to increase taxes by."

    Can you let me know the answer for #2.


    Thanks,
    Mitchell

  • game theory -

    Consider the following game.
    Player 2
    L R
    U 6, 1 8, 3
    Player 1 UM 4, 9 8, 4
    DM 7, 2 6, 9
    D 5, 4 9, 3
    (a) Is there a mixed strategy Nash equilibrium in which player 1 is placing positive probability only to strategies DM and D? If yes, what is the equilibrium? If no, show why?
    (b) Is there a mixed strategy Nash equilibrium in which player 1 is placing positive probability only to strategies U and DM? If yes, what is the equilibrium? If no, show why?
    (c) Report all Nash equilibrium (in pure or mixed strategies). For each equilibrium you report, compute mixing probabilities (if applicable) and verify that derived strategies are a Nash equilibrium.

    Consider the following game.

    Player 2
    L C R
    U 2,1 4,9 8, 2
    Player 1 UM 4, 9 5, 0 8, 4
    DM 5, 2 7, 3 6, 9
    D 5, 3 5, 4 9, 3
    (a) Identify all pure strategies that are strictly dominated by other pure strategies (in the entire game).
    (b) Can you find a pure strategy (not included in those you mention in your answer in part a) that is strictly dominated by a mixed strategy? Support your answer by applying the de finition of strict
    dominance and deriving the inequalities that must hold.
    (c) Is this game dominance solvable by iterated elimination of strictly dominated strategies? If yes, detail each round of elimination. If not, which strategies can be eliminated by iterated elimination of strictly dominated strategies?
    (d) Find all Nash equilibriumia in pure or mixed strategies

    Consider a three-player game in which players have two available strategies: to contribute or not to a public good. The public good is provided if at least two out of the three players choose to contribute. Each player gets a benefit t of 1 if the good is provided, and zero benefi t if the good is not provided. If a player chooses to contribute,
    she pays a cost c <1/2, and incurs zero cost if she chooses not to contribute. The final payoff is the net benefit (bene fit minus cost). Solve for a mixed strategy Nash equilibrium for this game such that players contribute with probability p and do not contribute with probability 1-p. (Hint: Given conjectured strategies, what is the probability that
    both out of two players contribute? What is the probability that exactly one out of two players contributes? Use these probabilities to
    compute players' expected payo ffs from each strategy and formulate
    players' indi fference condition).

    I can't get the player 1 and player two charts to look right.....but the player 1 U,UM,DM, D and player 2 is L,R and L, C, R. Numbers on the left of the comma is player1 actions and the player2 actions are on the comma.

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