# Optimal Strategies for the Clinton/Trump Debate

Consider modelling the Clinton/Trump debate via a static game in which each candidate can choose between two strategies: $\{A,P\}$, where $A$ denotes predominantly “attacking” the other candidate, while $P$ denotes predominantly discussing policy positions.

Further, let us consider the mixed strategies $\sigma_1 = (p,1-p)$ for Clinton, and $\sigma_2 = (q,1-q)$ for Trump. That is, Clinton predominantly attacks Trump with probability $p$, and Trump predominantly attacks Clinton with probability $q$.

Let us first deal with the general case of arbitrary payoffs, thus, generating the following payoff matrix:

$\left( \begin{array}{cc} \{a,b\} & \{c,d\} \\ \{e,f\} & \{g,h\} \\ \end{array} \right)$

That is, if Clinton attacks Trump and Trump attacks Clinton, the payoff to Clinton is $a$, while the payoff to Trump is $b$. If Clinton attacks Trump, and Trump ignores and discusses policy positions instead, the payoff to Clinton is $c$, while the payoff to trump is $d$. If Clinton discusses policy positions while Trump attacks, the payoff to Clinton is $e$, while the payoff to Trump is $f$, and if both candidates discuss policy positions instead of attacking each other, the payoff to them both will be $g$ and $h$ respectively.

With this information in hand, we can calculate the payoff to Clinton as:

$\pi_c(\sigma_1, \sigma_2) = a p q+c p (1-q)+e (1-p) q+g (1-p) (1-q)$

while the payoff to Trump is:

$\pi_t(\sigma_1,\sigma_2) = b p q+d p (1-q)+f (1-p) q+h (1-p) (1-q)$

With these payoff functions, we can compute each candidate’s best response to the other candidate by solving the following equations:

$\hat{\sigma}_1 \in \text{argmax}_{\sigma_1} \pi_1(\sigma_1,\sigma_2)$

$\hat{\sigma}_{2} \in \text{argmax}_{\sigma_2} \pi_2(\sigma_1,\sigma_2)$

where $\hat{\sigma}_{1,2}$ indicates the best response strategy to a fixed strategy for the other player.

Solving these equations, we obtain the following:

If

then,

Clinton’s best response is to choose $p = 1/2$.

If

then,

Clinton’s best response is to choose  $p = 1$.

Otherwise, her best response is to choose $p = 0$.

While for Trump, the best responses are computed as follows:

If

Trump’s best response is to choose $q = 1/2$.

If

Trump’s best response is to choose $q = 1$.

Otherwise, Trump’s best response is to choose $q = 0$.

To demonstrate this, let us work out an example. Assume (for this example) that the payoffs for each candidate are to sway independent voters / voters that have not made up their minds. Further, let us assume that these voters are more interested in policy positions, and will take attacks negatively. Obviously, this is not necessarily true, and we have solved the general case above. We are just using the following payoff matrix for demonstration purposes:

$\left( \begin{array}{cc} \{-1,-1\} & \{-1,1\} \\ \{1,-1\} & \{1,1\} \\ \end{array} \right)$

Using the above equations, we see that if $0 \leq q \leq 1$, Clinton’s best response is to choose $p=0$. While, if $0 \leq p \leq 1$, Trump’s best response is to choose $q =0$. That is, no matter what Trump’s strategy is, it is always Clinton’s best response to discuss policy positions. No matter what Clinton’s strategy is, it is always Trump’s best response to discuss policy positions as well. The two candidates’ payoff functions take the following form:

What this shows for example is that there is a Nash equilibrium of:

$(\sigma_1^{*}, \sigma_{2}^{*}) = (0,0)$.

The expected payoffs for each candidate are evidently

$\pi_c = \pi_t = 1$.

Let us work out an another example. This time, assume that if Clinton attacks Trump, she receives a payoff of $+1$, while if Trump attacks Clinton, he receives a payoff of $-1$. While, if Clinton discusses policy, while being attacked by Trump, she receives a payoff of $+1$, while Trump receives a payoff of $-1$. On the other hand, if Trump discusses policy while being attacked by Clinton, he receives a payoff $+1$, while Clinton receives a payoff of $-1$. If Clinton discusses policy, while Trump discusses policy, she receives a payoff of $+1$, while Trump receives a payoff of $-1$. The payoff matrix is evidently:

$\left( \begin{array}{cc} \{1,-1\} & \{1,-1\} \\ \{1,-1\} & \{1,-1\} \\ \end{array} \right)$

In this case, if $0 \leq q \leq 1$, then Clinton’s best response is to choose $p = 1/2$. While, if $0 \leq p \leq 1$, then Trump’s best response is to choose $q = 1/2$. The Nash equilibrium is evidently

$(\sigma_1^{*}, \sigma_{2}^{*}) = (1/2,1/2)$.

The expected payoffs for each candidate are evidently

$\pi_c = 1, \pi_t = -1$.

In this example,  even though it is the optimal strategy for each candidate to play a mixed strategy of 50% attack, 50% discuss policy, Clinton is expected to benefit, while Trump is expected to lose.

Let us also consider an example of where the audience is biased towards Trump. So, every time Trump attacks Clinton, he gains an additional point. Every time Trump discusses policy, while Clinton does the same he gains an additional point. While, if Clinton attacks while Trump discusses policy positions, she will lose a point, and he gains a point. Such a payoff matrix can be given by:

$\left( \begin{array}{cc} \{1,2\} & \{-1,1\} \\ \{0,1\} & \{0,1\} \\ \end{array} \right)$

Solving the equations above, we find that if $q = 1/2$, Clinton’s best response is to choose $p =1/2$. If $1/2 < q \leq 1$, Clinton’s best response is to choose $p = 1$. Otherwise, her best response is to choose $p = 0$. On the other hand, if $p = 0$, Trump’s best response is to choose $q = 1/2$. While, if $0 < p \leq 1$, Trump’s best response is to choose $q = 1$. Evidently, there is a single Nash equilibrium (as long as $1/2 < p \leq 1$):

$(\sigma_1^{*}, \sigma_{2}^{*}) = (1,1)$.

Therefore, in this situation, it is each candidate’s best strategy to attack one another. It is interesting that even in an audience that is heavily biased towards Trump, Clinton’s best strategy is still to attack 100% of the time.

The interested reader is invited to experiment with different scenarios using the general results derived above.