The Leaders Dilemma: How to Generate Cooperation at Home or Work

Motivating Joke

One I found in my email:

A couple in their nineties was having trouble remembering things so they went to their doctor for checkups. The doctor told them that they were both physically fine and advised them to write things down to help them remember.

Later that evening while watching television, the husband got up from his chair to go to the kitchen for a snack. He asked his wife if she wanted anything.

“Could you bring me a bowl of ice cream?” she asked.

“Sure,” he replied.

“Do you think you should write that down to remember it?” she asked.

“No, I can remember that,” he said.

“I’d like some strawberries on it, too. Do you need to write that down?” she said.

“No, I can remember that, too. Ice cream with strawberries,” he said, becoming a little irritated.

“I’d like some whipped cream on it, too. Can you remember all that? The doctor said you should write things down,” she said.

“For goodness sakes, I can remember that. I don’t need to write it down. A bowl of ice cream with strawberries and whipped cream,” he said, now more than a little irritated.

Off he went to the kitchen. About 20 minutes later he returned with a plate of bacon and eggs.

The wife stared at it for a moment and said, “Where’s my toast?”

The wife’s reaction is great. It’s clear that no matter what the husband brought, she was going to question him. This is in fact an issue that goes far beyond gender jokes.

It’s a problem about how hard it is to go first or be the leader. No matter what you do, someone critical will be sure you are wrong and inevitably figure out a flaw in your plan. The complaints increase with group size, and perhaps this is why groups make such bad decisions. It’s not the leader’s fault–it’s everyone stirring the pot.

The Defining Problem

Leaders often have to decide between great outcomes that might backfire and mediocre ones that work for sure. It’s what I call “The Leader’s Dilemma.”

The issue stems in large part because the game has a fixed order. Because the leader has to act first, followers have time to observe flaws and make criticisms. Often, the good outcomes need cooperation so they are risky and less likely to win out. In turn, safer but mediocre outcomes rise to the top.

Why can’t we all just get along?

Well, it turns out we sort of can. Here’s an idea: if sequential play is the main cause of the problem, why not change that feature? Why not change the game into simultaneous play where players have to guess and commit to actions? Perhaps if all players moved without full knowledge, they can all be made better off on average.

Indeed, this turns out to be a valuable option.

So let’s get to the details. I want to illustrate the idea through a game I created about two retailers competing on clothing products. One business moves first and is the “leader.” The other business is the “follower” and can threaten the good outcome with defection. What results in sequential play is a mediocre outcome for both businesses (there is another outcome with payoffs better for both parties–that is, the outcome is Pareto dominated).

The situation can be improved by changing the game into simultaneous play. In this case, the businesses have to guess at each others’ actions, so they will randomly commit to strategies. By shielding information, the leader business can produce a higher surplus for the group. On average, the surplus helps all involved parties. It’s a classic win-win.

Special thanks to Dimitris for a discussion inspiring this article.

The Game–Sequential Play (Full Knowledge of Action)

Imagine two businesses engaged in competition. The “leader” company moves first and announces whether it is creating a low-margin product (L) or a high-margin product (H). Upon observing the announcement, the “follower” business has two responses. It can either create an imitation product (I) under its own brand name, or it can just choose to match the product (M) by stocking it also.

You can think about the leader company as a brand name clothing retailer, and the follower company as a discount retailer, like Wal-Mart.

Assume the high-margin product has a market payoff of 4 million dollars, to be split between the two stores, and the low-margin product has one of 2 million dollars.

Whenever the follower store chooses to match (M), the stores split the profits evenly.

The interesting part is when the follower store imitates (I). If the follower store imitates the high-margin product, the knock-off will actually be good enough to capture all orders and gain the entire market. On the other hand, if the follower store imitates the low-margin product, the knock-off will be so low quality that no one will buy it, allowing the leader store to capture the entire market.

How does the game play out?

The Diagram

The payoffs are as follows:

The Outcome

Using backwards induction, we can figure out how the follower will respond. In the low-margin scenario, it will match (1) over imitate (0). In the high-margin scenario, the second retailer will imitate (4) instead of match (2).

The leader can anticipate choose accordingly. Picking the high-margin product leads to an imitation product with no payoff (0), and picking the low-margin product leads to a split and some payoff (1).

The leader will accordingly choose the low-margin product. Both stores get a payoff of 1 million dollars.

The Interpretation

The outcome is mediocre. Both stores end up with a payoff of 1 million dollars. If they instead could agree to split the high-margin product, they would both have 2 million dollars. This would be better for both sides.

While both sides would agree the outcome is better, there is a problem. There’s no way to enforce getting to that outcome. If the leader picks the high-margin product, the follower would not match and split, but it would choose to imitate and capture the entire market.

The two companies might try to write a contract to enforce cooperation, but I think market agreements of this sort are illegal.

If the game is repeated, say for hundreds of products, then both stores lose out big on every product.

How can they achieve a better outcome?

One option is to change the sequential play into simultaneous play.

The idea is to get the players to choose at the same time. This might be accomplished by hiring a mediator to collect strategy commitments. The second business needs to decide what it’s doing without knowing the first business’s decision.

The Revised Game–Simultaneous Play (No Knowledge of Action)

The game will illustrate the benefit to limiting knowledge through simultaneous play. Because neither knows exactly the other response, each will be forced to mix strategies. The result is higher average payouts—for both businesses. This is very useful if the game is repeated or played over several products.

Simultaneous play can be represented by a matrix:

The key feature is there is no pure strategy Nash equilibrium. If you try to figure out the best response, you’ll end up with a cycle:

– “low-margin” is best responded by “match”, which is best responded by “high-margin”, which is best responded by “imitate”, which is best responded by “low-margin”, and so on…

The situation is similar to the game of rock-paper-scissors. Every action is best responded by another action and players are never happy to stay put in one given outcome.

What’s the answer? Like rock-paper-scissors, both players need to randomize strategies.

How do they do that? They will randomly choose actions until the opponent is indifferent between the pure strategy choices. An illustration: when you pick rock-paper-scissors in equal frequency, your opponent will win 1/3 of the time for the pure strategies of rock, paper, or scissors.

In other games, the probability weights might not be equal. In this game, it turns out that player 1 will pick L with probability 2/3 (and H with probability 1/3) and player 2 will pick I with probability 1/3 (and M with probability 2/3).

I’ve included the equations at the end for completeness, as I won’t go through the math here.

When all is said and done, both players will mix between their choices. What this does is it allows for some of the higher payoff outcomes to be included for both players. Sometimes players 1 or 2 capture the market entirely, and sometimes player 1 and 2 split the high and low outcomes. They are not stuck in the one choice of splitting the low-margin market.

As a consequence, the payoff to each player increases 33 percent from 1 million dollars to 1.33 million dollars.

I guess it can be better not to know. Ignorance can be business bliss.

Conclusion

There are other times where people try to change games from sequential to simultaneous play.

Here’s one example for couples: rather than have a back and forth argument, counselors try get couples to write down their thoughts on paper separately an independently. This is one way for each person to act alone without knowledge of what the other person will act.

Add some randomness. It’s the spice of life.

Appendix: Mixed Equilibrium Math

Let’s assume player 1 chooses L with probability p and H with probability 1-p. The probability will be set exactly so that player 2 is indifferent between the choices of I and M.

Now assume player 2 chooses I with probability q and M with probability 1-q. The probability will be set exactly so that player 1 is indifferent between the choices of L and H.