For those who just joined us, we're looking at an imaginary problem where a newspaper is trying to incentivize subscribers to vote on beauty contest entrants, and we're trying to work out the best incentive structure.
In Part 2, we showed that paying a flat fee for every vote may lead to random voting; in this installment, we'll take a look at doing something a little bit more sophisticated. We'll also introduce a notion of Nash Equilibrium, and show that the question of paying people to vote may actually lead to some weird behavior.
The problem with the first approach was that the incentive was not correlated to the quality of the vote. What we'd really like to do is pay people who submit "good" votes more than we pay people who submit "bad" votes. The problem is that a priori (that's economist-speak for "before we start"), we can't tell the difference between good and bad votes.
Let's try something that sounds clever: let's collect all the votes, find the winning beauty contest entrant, and give $10 to everybody whose vote was for the winner. This is starting to look reminiscent of Steem! It turns out that this is called a "Keynesian Beauty Contest." Yes, that economist with the weird lips (that everybody around here loves, I'm told), John Maynard Keynes, came up with this analogy between the stock market and a beauty contest in which people are paid money if their vote turns out to be for the winner. I'll let people debate the validity of the analogy in the comments to this article; we'll just stick with the beauty contest here.
I'll build a concrete example. To keep things simple, let's have voters choosing between two beauty contest entrants:
Aww, how cute! A puppy (P) and a kitten (K)! So the first two components of our game specification are in hand already: we have players (voters, I'll also call them agents), and we have actions (vote P or K). To fully specify a game, we need to define the 3rd part: payoffs.
Our payoff functions will have two components. The first is "idiosyncratic": each person has a personal preference for P or K, and in principle the strength of each person's preference could be different. So we'll use the letter vi to denote this preference (v stands for "value"). If voter i prefers P, then vi represents how much voter i wants to see P win (alternately, if i prefers K, then that's the strength of i's K preference). Think of vi as the maximum amount of money that i would pay to change the result from K to P. Of course, the second payoff component is the monetary reward for picking the winner; just like in Part 2 we'll call this p.
Now we have all we need to define the utility function for agent i. We'll write ai to denote agent i's vote, and a-i to denote everybody else's vote. If agent i is a puppy-lover, here is the utility (payoff) function:
Just so we're absolutely sure we know what we're doing, let me pick that apart line-by-line:
- If my favorite wins and I voted for it, I get paid and get happy.
- If my favorite wins but I didn't vote for it, I don't get paid but at least I'm still happy that my favorite won.
- If my favorite loses, but I voted for the winner, I'm not happy but at least I get paid.
- If my favorite loses, but I voted for it anyway, I'm neither happy nor paid.
See where I'm going with this? It looks like paying the winners could actually be setting up a perverse incentive for people to lie about their favorite. Let's analyze this formally.
First, let's assume that there are a lot of voters, and the margin of victory is at least 2 votes, so that no single voter can switch their vote and change the outcome. This will usually be true, and it helps focus on the issue of perverse incentives. Before I go any further, let's chew on the following definition:
Definition: Nash Equilibrium. A Nash Equilibrium (NE) is a collection of votes in which no voter wishes they could individually switch their vote to something else.
This is the intuition: First, we let everybody vote. Then we go up to each voter in private, show them the result, and ask them if they would be better off having voted for something different. If every person says "nope, I'm happy with how I voted," then we call that a Nash Equilibrium. With this definition in hand, I'm going to make a claim:
Claim: When p > 0, regardless of how strong peoples' individual preferences are, there are exactly two Nash Equilibria: one where everybody votes P, and one where everybody votes K.
How can this be true? What we're saying is that I could like puppies SO MUCH - my vi for puppies could be a billion - but still I'd prefer to vote for kittens. Why? We need to look at the payoff function to understand it! Suppose kittens win the contest. That means we're living in the last two rows of the payoff function. Well, that makes it simple! I could stick to my guns and vote P, but then I'd get a payoff of 0. Or, I could write down K instead, vote with the crowd, and I'd get paid money! Notice that vi does not show up in the last two rows; if puppies don't win, it doesn't make any difference how much I like them. I have a strong incentive to vote for kittens, even though I actually want them to lose.
Now let's bring this back to Steem. Steem works by paying people to vote for the winners, just like in the kittens/puppies game. At first glance, this seems to make sense - the winners should be the highest-quality, right? But the story of puppies and kittens highlights a subtle issue: paying people to vote for winners can skew the incentive away from voting for quality, and towards voting with the crowd and participating in groupthink.
By all means, the Steem system is a lot more complex than our toy puppies/kittens example, so the story isn't done yet. We have a lot left to analyze! I haven't even begun to talk about short-term versus long-term payoffs, and we haven't considered the fact that voters in Steem have multiple votes, and I haven't even touched on the question of what incentives the content-creators experience.
Also, let me give credit where credit is due: I brushed up on this material by reading this article and looking through these lecture slides by Muriel Niederle at Stanford University. I also just ran across this related article by @smooth: Voting is a popularity contest.
My other articles you may find interesting:
Part 1 of the game theory series: Introduction
Part 2 of the game theory series: Beauty Contests
And now, before I go, I'll pose a discussion question: Why should we be skeptical of my simplified puppies/kittens example?