The Colonel Blotto problem is a zero-sum game about how to best position resources. While Colonel Blotto games are described in a military context, I will explain in future articles some of its useful applications in sports, advertising, elections, and many other areas.

Today I want to highlight a specific Colonel Blotto game that is convenient to solve mathematically.

A simple example of a Colonel Blotto game

Colonel Blotto is planning his attack for the next day. He wishes to capture 2 different locations, and he has 4 regiments of troops.

The opposing commander, Colonel Lotso, also wishes to attack those 2 locations, and he has 3 regiments.

To make the problem precise, it is necessary to ascribe payoffs to possible outcomes.

Let’s say that if both send troops to the same site, the person who sent more troops will win the site.

Specifically, if one Colonel sends x troops, and other sends a lesser number y, then the Colonel who sent x regiments will win. The payout will be equal to y + 1: the Colonel gets y points for defeating that number of enemy troops, plus he gets 1 point for securing the location.

For example, if Colonel Blotto sent all 4 troops to one site, and Colonel Lotso all 3 to the same site, then Colonel Blotto would win the battle and get 4 points because he defeated 3 of Lotso’s regiments and got 1 point for securing the location.

If each sent all of their troops to different sites, however, then both would gain a location but not defeat any enemy troops. Each Colonel would get a location, but neither gains an advantage, so the net payout to each would be 0.

If both Colonels play strategically, how should they distribute their troops?

Solution to the Colonel Blotto game

Colonel Blotto has 5 strategies at his disposal: he can send 4 troops to either location, he can send 3 to one location and 1 to another, or he can send an equal number of troops 2 to each location. We will abbreviate these in set notation (4, 0), (0, 4), (3, 1), (1, 3), and (2,2).

By similar reasoning, Colonel Lotso has 4 strategies at his disposal that can be abbreviated (3, 0), (0, 3), (2, 1), and (1, 2).

We will write out the 5 x 4 matrix for the set of strategies and diligently calculate the payout for each battle. Since this is a zero sum game, we will write the payouts in terms of Colonel Blotto, and note that Colonel Lotso’s matrix would have the opposite values.

(The matrix displays net payouts to each site, so careful accounting is required to calculate some of the payouts. For example, consider the strategy (4,0) versus (2,1). Blotto wins the first site and gets a total of 3 points. But Lotso wins the other site and gets 1. So the net payout to Blotto is 3-1 = 2).

We can use a handy zero gum game solver to figure out the optimal strategy.

Colonel Blotto will play (4,0) with probability 4/9, he plays (0, 4) with probability 4/9, and he will play (2,2) with probability 1/9. He never plays (3, 1) or (1, 3).

Colonel Lotso does not have a unique mixed strategy. He can actually play a variety of mixed strategies. The symmetric solution, as explained in Introduction to game theory, is that he will play both (3,0) and (0,3) with probability 1/18, and he will play both (2,1) and (1,2) with probability 4/9.

Interpreting the solution

When both are playing their optimal strategy, Blotto can expect a payout of 14/9. This makes sense that Blotto has a positive payout because he has 1 extra regiment and should be favored.

The solution indicates that Blotto should concentrate his troops to specific sites and occasionally split his troops, just to make sure that Lotso cannot steal a location easily.

Lotso responds by doing the opposite. Lotso cannot win against Blotto in the numbers game, so Lotso has to spread his troops out and hope to secure an undefended location with 1 regiment. Occasionally Lotso will deploy all his troops to one site or the other, just so that Blotto cannot win by spreading his troops evenly all the time.

It’s not exactly a scientific observation, but time and again, I have seen friends whipped by crazy girlfriends. (I would equally say there are a lot of crazy boyfriends, so don’t get caught up in the gender).

It turns out there is a game theory explanation for this phenomenon which is what I want to explore in this article.

You’ll see why crazy people get their way, and how you can use a similar strategy to fight back.

Battle of the sexes

To begin, consider the following model of conflict. This game is one of the simplest examples in game theory, often the first example presented in a game theory course.

Alice and Bob are planning to go out for an evening. Bob wants to go to the football game, but Alice wants to go to the opera. Still, both would prefer to be with the other person than going out to an event by themselves.

Let’s say each person gets 3 points to go to their favored event, 1 points to go to the other event but be with their significant other, and 0 points if the two do not go together.

The outcomes can be represented in the following matrix:

The question is: how will this game play out?

Solution 1: opera or football

This game is very easy to solve. We need to consider what each person’s best response is, given what the other person might be doing.

Here is how Bob thinks about the problem:

–If Alice goes to the football game, then I should also go to get 3 points rather than going to the opera for 0 points

–If Alice goes to the opera, I might not like the opera, but if I go I’ll at least get 1 point. If I watch football I would instead end up with 0 points.

Bob’s strategy is to guess what Alice wants to do and follow.

Alice’s reasoning is exactly the same: she wants to be with Bob, so she should pick the choice she expects him to do.

The outcomes that both go to football, or both go to opera are the solutions of the game.

These are the Nash equilibria of the game.

The pleasant finding is that both players end up choosing mutually beneficial outcomes.

But the annoying part is the game has two different solutions. I mean which one do they end up doing?? Do they go to football or opera? The concept of the Nash equilibrium falls short here in providing a specific prediction.

However, there are other ways to think about the game.

Solution 2: correlated equilibrium

If the game is repeated, as it would be in a relationship, there is a good compromise that Alice and Bob could agree to.

What they could do is flip a coin to decide whether they go to football or the opera. Over time the events will balance out, and they both guarantee that they are together. They will get an average payout of 2.

This seems like a very reasonable solution. It also makes a lot more sense they flip a single coin and coordinate their choices, rather than each player flipping a coin individually and hoping to end up at the same place (the mixed strategy Nash equilibrium doesn’t make much sense in this game: a good chunk of time the two end up in different places)

In a healthy relationship, there is compromise and people may choose the correlated equilibrium.

But a crazy girlfriend would not tolerate this. She wants to get her way, and she uses another strategy.

Solution 3: changing the game

Let’s suppose Alice really, really wants to go to the opera, and she wants to convince Bob that it’s in his best interest too.

Here is a drastic strategy that Alice could use to change the game.

Alice takes out a $20 bill from her purse, and then announces the following:

Look Bob, I really want to go to the opera. And if I think there is even a chance we are not going, that would stress me out. I’ll vent by burning this $20 bill.

I’m going to the room to think about what I’m going to do. Come knock on my door with your choice in one minute. But think about what’s best for both of us.

Alice has introduced another strategy to the game called burning money. In this game, Alice first chooses whether to burn her own $20 bill, and then both players choose where they would like to go.

Let’s say that burning a $20 bill will destroy 1 point of utility for Alice. How will this game play out?

The way to analyze this game is to write out a large matrix with all the choices and solve as one normally would.

Alice has two moves in her strategy: she can either burn the money or not, and she can either go to the opera or not. Let’s abbreviate Alice’s choice of “burn the money, go to opera” as BO, and similarly the rest of her choices as BF, NO, NF.

Bob also has two moves in his strategy. He needs to decide what he will do if Alice burns the money, and what he will do if Alice does not burn the money. Let’s abbreviate Bob’s choice of “if she burns the money I go to the opera, if she does not burn I go to football” as OF, and similarly the rest of his choices as OO, FO, FF.

The matrix of payouts is as follows:

We can now eliminate bad strategies by a concept of iteratively deleting weakly dominated strategies (see more in this game and this game).

To begin, Alice will realize that NO is weakly better than BF, so she will never play the strategy of burning the money and choosing football. Both players realize this and “cross out” that option in the matrix:

Looking at the matrix, Bob realizes that OO is weakly better than OF, and FO is weakly better than FF. So both Alice and Bob eliminate those options in their mind.

The process continues, and you can check the only strategies that remains are NO for Alice and OO for Bob.

The result is this: the equilibrium outcome is that Alice does not burn the money and goes to the opera, and Bob chooses to go to the opera whether Alice burns money or not.

That is, they end up going to the opera just like Alice wanted!

Just think about what happened: Alice got her way because she threatened to burn money. But she never actually has to burn the money: she gets her way because she threatens to torch her own utility.

This seems to capture an element of how spoiled brats in real life operate. They do not always throw tantrums. They only have to threaten to throw a tantrum and act unhappy to force everyone into their choice.

(While I find the solution interesting, I should mention there is controversy about the idea. It is odd that Alice can change the game by threatening to use bizarre behavior. This is an issue raised in this paper )

How to fight back

Bob has a couple of options for fighting back in this game. He can threaten to burn money pre-emptively too, which might get Alice to see his side and drop the pettiness.

Or he can play it safe and change the game once and for all. While crazy people do change and grow, it is a question of how fast and whether it is worth the effort.

Of course there are other ways to deal with crazy people, and I could go on and on. But in my opinion it is often not worth the time and effort. Sometimes breaking it off is the best move.

Let’s you and I play this very simple game and analyze the best strategy.

Imagine we are playing this game in a college experiment. We each have a chance to win money depending on how we play.

Here are the rules:

–You and I each secretly play “A” or “B”

–If we both pick “A,” then we each get $4

–If one person picks “A,” and the other “B”, the person picking “A” gets $1 and the person playing “B” gets $3

–If we both pick “B,” then we each get no money and leave with $0

Here is the matrix of payouts:

The game is played once. What option would you pick?

Analyzing the game

This game is a no-brainer: it is a dominant strategy to pick “A” and both of us should get $4.

Verifying this is an easy task. Each person thinks about the best response to the other player’s move. If the other player picks “A,” then it’s best to also pick “A” to get $4 rather than “B” to get $3. If the other player picks “B,” it is also better to pick “A” and get $1 rather than “B” to get $0.

The best strategy is to pick “A,” regardless of what the other person is doing. Both players should easily cooperate and get $4.

There is no sensible reason to pick “B.” And yet, that’s exactly what researchers found people doing over half a century ago, in a similar game played with pennies rather than dollars.

The results were astounding: more than 50 percent ended up playing the strategy “B”!

(The experiment is referenced in The Survival Game regarding this 1960 article)

We could be tempted to chalk up the result to the small payouts, or maybe people did not understand the rules. It is possible that people did not take the game seriously.

But the researchers also raised another possible, biological explanation that’s worth investigating.

The green-eyed monster of jealousy

As explained above, both players maximize their payout when they pick “A.” It should be obvious that picking “A” is the best thing to do. Except, perhaps we are thinking about the problem with the wrong motivation.

In game theory, economics, or business, we often choose the option that brings us the highest profit in absolute terms. All things equal, we would rather have $1,000 than $100.

But people do not always think in absolute success. They can sometimes think in terms of relative success: the goal is not to maximize payout, but rather, in the researchers’ words, “to maximize the difference between one’s self and the other player.”

There is further experimental evidence of this idea, as explained on Michael Shermer’s blog:

Would you rather earn $50,000 a year while other people make $25,000, or would you rather earn $100,000 a year while other people get $250,000? Assume for the moment that prices of goods and services will stay the same.

Surprisingly — stunningly, in fact — research shows that the majority of people select the first option; they would rather make twice as much as others even if that meant earning half as much as they could otherwise have. How irrational is that?

…In this case, relative social ranking trumps absolute financial status.[emphasis mine]

The point is that relative thinking, and jealousy of other people’s success, factors into how people think about money.

So let’s use this idea to re-analyze the game.

The destructive nature of relative thinking

The original payout matrix accurately reflected the total payouts for each player. It was apparent that “A” was the obvious choice.

But imagine that people were thinking in terms of relative rather than absolute payout. That is, they only cared about how much more they earned than the other player.

The game is now transformed into the following payouts:

–If we both pick “A,” then we each get $0 more than the other person

–If one person picks “A,” and the other “B”, the person picking “B” gets $2 more than the other player

–If we both pick “B,” then we each get no money and also leave with $0 more than the other person

Here is the matrix of relative payouts:

The payouts of this game are completely changed from the original one. Notice how the original game–which was non-zero sum and mutually profitable–is now suddenly a zero sum, and competitive, game.

The strategy becomes competitive too: in this game, it is a dominant strategy to pick “B,” meaning both players are expected to leave with nothing.

Rather than cooperating for mutual gain, both players “happily” end up with nothing to avoid letting the other person gain in stride.

Get over your money jealousy

This outcome is sadly not just theoretical: people gleefully act towards mutual destruction out of money jealousy.

I went over many such examples in a previous article. In that article, I made a plea for people to be more calm and not worry so much about other people’s success.

Here’s my closing advice from that article. I hope it will inspire people to be less jealous and look for mutual gain:

I have my own personal analogy to get over jealousy. I think about success as filling up water flowing from an ocean. Each of us has a different size glass that represents a personal level of achievement. There’s really no point worrying if your neighbor has a bigger glass than you since there is more than enough water to go around. If you want to get more, then focus on what you can do. Success will come from building your own glass and filling it, not from shattering what your neighbor has. It’s time to put the green eyed monster of jealousy to rest.

Optimal strategy in spinning the wheel

Strategy for Pay the Rent

Strictly dominated strategies in Lucky Seven

I came across another instance of game theory in a British TV show called Golden Balls. One of the games in the show is called “Split or Steal” and it closely resembles the classic game of the Prisoner’s Dilemma.

The twist in the game is that players are in the same room and can communicate, which yields some entertaining dialog and reactions.

Below is the video where one person decides whether to trust his partner with £100,000 on the line.

Video: Golden Balls – £100,000 Split Or Steal?

Rough transcript of the show

Host: This is serious, life-changing money. The jackpot today is £100,150. You have one final decision to make. You are going to play “Split or Steal.” I know you’re the last two people in the country I have to explain this to, but you have two final golden balls.

You each have a golden ball with the word “Split” written inside. You each have a golden ball with the word “Steal” written inside. You will make a conscious choice choosing the “Split” or the “Steal” ball.

–If you both choose the “Split” ball, then you split the jackpot of £100,150 and go home £50,075 richer.

–If one of you “Splits” and one of you “Steals,” then whoever chooses the “Steal” ball will go home with £100,150. The person who chooses the “Split” ball goes home with nothing.

–If both of you choose the “Steal” ball, then both of you go home with nothing.

[Note: the game resembles a Prisoner's Dilemma, with payoffs as follows (via Wikipedia):

Result	Split		Steal
Split	50%	50%	100%	0%
Steal	0%	100%	0%	0%

The game is a version of the Prisoner's Dilemma: it is in each player's interest to steal the jackpot.

That is, you want to trust your partner and pick split, but if you know your partner will split, then you would rather steal and take everything for yourself.

The outcome is that both players are tempted to steal, and if they do that, then both end up with nothing.

Let's see how the contestants fare on the show, and whether they can cooperate.]

Before I ask you to choose, I want you to look at your golden balls and make sure you know which is the “Split” and which is the “Steal” ball. This is very important, and make sure you don’t show it to us.

Before I ask you to choose, I think you have some talking to do to each other.

[Important background note: Both Stephen and Sarah are returning contestants from previous games who all "Split" where their opponent "Stole."]

Sarah: Stephen, I just thought they weren’t puppy dog tears and they were real tears, and that you were genuinely going to split that?

Stephen: I am going to split this. 50,000, I’m just, it’s unbelievable. I’m very, very happy to go home with 50,000.

Sarah: You’re telling me you are going to split?

Stephen: If I stole off of you, every single person would come over here and lynch me.

Sarah: There’s no way I could. I mean everyone who knew me would just be disgusted if I stole the money.

Stephen: When people watch this, they are not going to believe it. Sarah, I can look you straight in the eyes and tell you I am going to split this money. I swear to you.

Sarah: That’s great.

Host: This is serious money. Sarah, Steve, choose either the “Split” or the “Steal” ball now. Hold it up.

Stephen: We are going home with 50,000 each. I promise you that.

Host: Split or Steal?

Stephen holds up SPLIT
Sarah holds up STEAL

Host: You never know what’s coming up in this game. Congratulations Sarah, you have just won £100,150. Stephen, I am so sorry, commiserations. You have just lost. So an unfamiliar feeling for one of you, but a horribly familiar feeling for another.

Stephen: Golden Balls has taught me that some people look for revenge quite easily. And greed knows no bounds.

Sarah: When I saw Stephen hold up the “Split” ball, I wasn’t proud, I wasn’t happy about what I had done. But having been stabbed in the back last time, I just couldn’t put myself through that again.

It’s a bit painful to see how the game turns out. Stephen tried his best, but ultimately he faced a no-win situation because Sarah could not overcome having been betrayed in a previous game. Sarah was clearly not playing the game at hand but instead considering a bigger game in which she wanted to avoid being double-crossed again.

It would have been nice to see them cooperate, but the true nature of the game is not always pretty.

The problem not only is about strategy, but its proof is interesting mathematically too. Here is the puzzle:

A traveler gets lost on a deserted island and finds himself surrounded by a group of n cannibals.

Each cannibal wants to eat the traveler but, as each knows, there is a risk. A cannibal that attacks and eats the traveler would become tired and defenseless. After he eats, he would become an easy target for another cannibal (who would also become tired and defenseless after eating).

The cannibals are all hungry, but they cannot trust each other to cooperate. The cannibals happen to be well versed in game theory, so they will think before making a move.

Does the nearest cannibal, or any cannibal in the group, devour the lost traveler?

As usual, I have given a solution in the comments section.

Can you figure it out?

(Also, after writing this post I found the same puzzle appeared on the Incidental Economist with a nice explanation too)