# Expected duration of a conflict

I'm currently struggling with some statistical work, and thus I'm seeking your advice. I'm trying to investigate the expected duration of a conflict in a) a democracy and b) an autocracy.

What I do know:

• the probability of onset of a conflict is:
• in democracies: 1%
• in autocracies: 2.6%
• the probability of continuation of the conflict from the year before (t-1)
• in democracies: 83%
• in autocracies: 86.1%

Thus, my question is, how do I calculate, how much time a conflict lasts for once started?

• You have to read on Survival and Hazard functions. e.g. rpubs.com/daspringate/survival May 31, 2017 at 15:47
• What about conflicts causing the countries to become autocracies? May 31, 2017 at 21:25

Given that a conflict has already started, the information about the onset of a conflict does not add to our problem. Depending on your measure of centrality, we may either proceed in any given way. I'd consider estimating the 50% mark on the survival function, that is, the median.

Now, the survival function for a conflict is the function,

$$S(t) = \begin{cases} 0.83^t & \text{in democracies} \\ 0.861^t & \text{in autocracies} \end{cases}, \qquad t \in \{0, 1, 2, ...\}$$

Where we may recognise a geometric distribution (I leave it to you to show that from $S(t) = 1 - F(t)$, for $F(t)$ a geometric distribution obtains). Then, for the median we find $.5 = .83^t \implies t = \frac{\log .5}{\log .83} \approx 3.72$ years in case of a democracy. Giving this an economic interpretation, we expect that 50% of conflicts last at most 3.72 years, assuming ceteris paribus.

In case of an autocracy, we similarly find $.5 = .861^t \implies t=\frac{\log .5}{\log .861} \approx 4.63$ years. So, we find that 50% of conflicts are expected to last at most 4.63 years, again assuming ceteris paribus.

For the expected value, we may proceed in two ways. There is a well-known result which states that the expected survival time $T$ may be expressed as the integral of the survival function, which in the discrete case reduces to,

$$E[T] = \sum_{t=0}^\infty S(t)$$

Doing so yields an expected conflict time of 5.88 years in the case of a democracy, ceteris paribus. For an autocracy, we similarly obtain 7.19 years.

We could alternatively have used our knowledge of the geometric distribution (like in @Aksakal's rule of thumb) to obtain the following,

$$E[T] = \frac{1}{1 - p}$$

Where $p =.83$ in the case of a democracy, yielding again 5.88 years. For an autocracy ($p=.861$), we similarly find 7.19.

The rule of a thumb is $1/r$ years where $r$ is the survival rate, in this case it's $\approx\frac{1}{1-0.83}$