We all know it happens–the underdog wins, the smaller army emerges victorious, the startup overtakes the multinational.

But why?

And how can we apply this effect to our companies, sports teams, or lives?

Modeling unequal conflict

Enter Colonel Blotto.

Colonel Blotto is a simple game, well-studied in game theory. The Colonel must choose how to distribute troops over some number of battlefields. The goal is to emerge victorious at more battlefields, but the Colonel doesn’t know in advance how many opponents will arrive at a given field. Victory is determined simply by allocating more troops at a given battlefield.

Fighting and winning at a disadvantage

Unfortunately, our Colonel is at a disadvantage and doesn’t have as many troops as the enemy.

For example, if we have 3 battlefields:

Despite this disadvantage, Blotto can still succeed:

This is our first bit of insight: picking your battles (and picking which to sacrifice) is unquestionably more effective given limited resources. This is hardly earth-shaking news, but it is good to see that our model bears out conventional wisdom.

In fact, there is a common application of this in politics: gerrymandering. By carefully choosing districts, politicians have learned to win elections without the popular vote.

Now, the obvious objection is that an Enemy with superior resources could easily reallocate troops to guarantee victory on a majority of battlefields—this is unfortunate for Blotto, but it reveals a second insight: the Enemy doesn’t have enough resources to win at everything! Unless the Enemy has a truly overwhelming number of resources, Blotto has a good shot at winning at least some battlefields, just as a small business can often outperform larger competitors in a few key areas by focusing on filling needs unmet by those larger competitors.

Fighting at a major disadvantage

But what can an outgunned Blotto do in a situation where the enemy has many more resources?

Even if our good Colonel allocates all troops to one battlefield, the Enemy has enough resources to prevent a Blotto victory on either battlefield.

So why not add more battlefields?

By adding new battlefields to the game, Blotto is able to be victorious on one battlefield despite having half the troops.

Of course this isn’t allowed in the game theory version (and Blotto still hasn’t won!), but it uncovers a third insight: adding new areas to compete in is advantageous for the underdog. Doing one thing (or a few things) well is a better strategy for small companies, and it can be a great advantage in a career as well.

This applies even more in situations where a victory on one battlefield may be sufficient to win–imagine a small company suddenly competing on grounds no other company knew existed.

The math behind the maxim

We have seen a few good insights suggested by Colonel Blotto that can be rephrased as more conventional wisdom:

1. Pick your battles. If you spread your resources too thin, you are easily defeated.
2. Know thy enemy. If you know how your opponents will use their resources, you can pick the right battles.
3. The underdog can win too. It’s easy to get disheartened, but remember that you can win even with fewer resources.
4. If you can’t win, cheat. Add a new dimension to the game, a new product, a new battlefield. It might just be the edge you need.