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It is known that the mixed strategy (\%$, \%$) is the only mixed Nash equilibrium for this game. So I supposed that Player 1 randomize to play $H$ with probability $p$ and Player 2 randomize to play $H$ with probability $q$. However, when $p=0$ player 2 should clearly choose $q=1$, and likewise when $p=1$ player 2 should choose $q=0$; in other words, neither of these situations is "stable".

Hence, \begin \hline Player1\backslash Player2 & H\,\text\,q & T\,\text\,(1-q) \ \hline H\,\text\,p & ( 1, -1) & (-1, 1) \ \hline T\,\text\,(1-p) & (-1, 1) & ( 1, -1) \ \hline \end I tried to do this but I cannot find the mixed Nash equilibrium (\%$, \%$). \begin \hline Player1\backslash Player2 & H\,\text\,q & T\,\text\,(1-q) \ \hline H\,\text\,p & (\color, -pq) & (p(q-1), \color) \ \hline T\,\text\,(1-p) & ((p-1)q, \color) & (\color, (1-p)(q-1)) \ \hline \end Player 1 chooses $p$ to make Player 2 not care about what value $q$ he chooses. More formally, this shows that when $q \neq 1/2$, the profile cannot be a Nash equilibrium. So $p=1/2$, $q=1/2$ is the only possible equilibrium.

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The matching pennies game is the following: \begin \hline Player1\backslash Player2 & H & T \ \hline H & (\color, -1) & (-1, \color) \ \hline T & (-1, \color) & (\color, -1) \ \hline \end Here, there is no pure Nash equilibrium in this game. However, when players are allowed to use mixed strategy, at least one Nash equilibrium is guaranteed to exist. 1/2$, $(1-2q)$ is negative so player 1 maximizes her expected utility by choosing $p=1$.

Then we discuss why we might be interested in Nash equilibrium and how we might find Nash equilibrium in various games.

As an example, we play a class investment game to illustrate that there can be many equilibria in social settings, and that societies can fail to coordinate at all or may coordinate on a bad equilibrium.

They put both of you in separate rooms (which is the non-cooperative part of the game).

If you snitch on Harry and he doesn't snitch on you, you get off with 1 year and he will receive 7 years. However, if you both end up snitching, you will both receive 4 years (see Table 1 below). Prisoner's Dilemma Let's examine all the possible scenarios here.

Then her friends will also turn them down because they wouldn't want to play second fiddle.

One of the most famous examples of a Nash equilibrium is the Prisoner's Dilemma, which explains how two criminals who are being interrogated separately for the same crime are better off confessing than remaining silent, as neither knows what the other will do. We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities.

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We argue that coordination problems are common in the real world.