3
$\begingroup$

We have to answer this question and I think I have done part (a) right but get stuck at part (b). Since $-0.5 \le \varepsilon_i \le 0.5 \ \forall i$, I seem to get a solution of the NE being TR, which does not seem right. Please could you help?

John Harsayni showed that a mixed strategy equilibrium of a perfect information game can be thought of as an approximation to an equilibrium (i.e. Bayesian NE) of a game where each player has a slight amount of incomplete information about the exact preferences of the other players. Consider the following Bayesian game, with payoffs as given below:

$\begin{array}{|c|c|c|} \hline & L & R \\ \hline T & 1 + \varepsilon_1, \varepsilon_2 & \varepsilon_1, 1 \\ \hline B & 0, 2 + \varepsilon_2 & 2, 0 \\ \hline \end{array}$

Nature chooses $\varepsilon_1$ and $\varepsilon_2$ independently, and both of these are uniformly distributed on the interval $[􀀀-k, k]$ where $k\lt 0.5$: Player $i$ is informed of the realization of $\varepsilon_i$; but not of $\varepsilon_j$ for all $j\neq i$: Players then choose actions simultaneously.

$(a)$ Solve for an Nash equilibrium of this game when $k = 0$:

$(b)$ Solve for a Bayesian NE when $k \gt 0$; $k \lt 0.5$: What is the probability assigned by player $i$ to the event that his opponent plays his first action in this BNE?

$(c)$ What do the probabilities in $(b)$ converge to when $k\to 0$?

Interpret your results.

I have attempted (a) as when $k=0$, there is no range for $\varepsilon_1$ or $\varepsilon_2$ to be on, so they must both be $0$ too, and then one can find mixed strategies for the game. But I am unsure how to introduce probabilities or to draw the tree for the Bayesian game in part (b). Please can you help?

$\endgroup$
  • 1
    $\begingroup$ Welcome to MathSE. You have posted a series of homework questions without showing any of your own work. We are not here to do your homework for you. Please put in some effort. $\endgroup$ – Théophile Dec 2 '15 at 15:10
  • $\begingroup$ I apologise. Yesterday was my first day of using stackexchange. I did not know how it worked in that you needed to show your efforts. I have edited and included some of my attempts on the questions but get stuck at part (b). Please could you help? $\endgroup$ – captaincook Dec 3 '15 at 11:46
  • $\begingroup$ Note that there is a parasitic Unicode character in the closed interval $[-k,k]$ under the table, that does not display properly (on my monitor, at least). Better remove it or replace it with the equivalent LaTeX code. $\endgroup$ – Alex M. Dec 3 '15 at 14:03
0
$\begingroup$

(a) If $k = 0$ , then $\varepsilon_1 = \varepsilon_2 = 0$, and both players know this. Using the normal method of finding mixed strategy which makes each player indifferent between playing either of their actions, we have player 1 playing $T$ with probability $\dfrac 2 3$ and $B$ with $\dfrac 1 3$, and player 2 playing $L$ with probability $\dfrac 2 3$ and $R$ with $\dfrac 1 3$.

(b) If $0 < k < 0.5$, it is still not possible for a PSNE to exist, so you will have to maximize each player's utility to find the BNE.

Computing it will give us player 1 playing $T$ with probability $\dfrac {2 + \varepsilon_2} 3$ and player 2 playing $L$ with probability $\dfrac {2 - \varepsilon_1} 3$. Note the magnitudes on the $\varepsilon$-terms differ between the players.

(c) As $k$ approaches $0$, $\varepsilon$ approaches $0$ as well, hence the mixed strategy NE in (a) can be used to approximate the BNE when $k$ is small.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.