Monday, 19 October 2009

Bird Calls: Honesty and Variation in Cheap Signalling Systems

The best poker players can read other people like blogs. They know when you're bluffing and they know when you've got the cards to back your bet. A lucky player may be able to fool some of the people some of the time, but not all of the people all of the time. Given the continual possibility of bluffing, however, why do players pay any attention to other people's bets? Why do players strive to earn respect?

Let's play some Hold 'em. Imagine that a group of people are playing poker. At each round, they are dealt a hand. They know the strength of this hand on its own but that's not the whole story - they must figure out how good their hand is compared to everybody else. Even a pair of twos wins sometimes. If you think you have a good chance of winning, you'll put a bet down. This is your threat display - your signal to show you have a good hand. Some people will respect this and retreat without a fight. Others, however, may judge their hand to be better, and challenge you with a call. The cards play out, the winner is revealed and tops up his stack with the loser's chips.

The trick is to put out a bet that is proportionate to your chances of winning. Too small and lesser players may choose to challenge you and they might win by chance. So, there's no point in underplaying (unless you're trying to draw people into a trap, but this has its dangers). However, what stops people from putting too big of a bet? To start with, people may think you're bluffing, but a big enough risk will usually scare most people off. Too much of this behaviour, however, and the bets become meaningless. I've seen many people play their first hand of poker against hardened pros and win, mainly because the newbies don't respect the relationship between bet and threat. By devaluing the signaling currency, they make the system of betting irrelevant, forcing the pros to play more by chance, leveling the field. After a while, though, the newbies will harden and fall in line. The big question is why.

The problem of honesty in a poker game is analogous to the same problem in competing animals. Some animals, such as sparrows, have ways of signaling their fighting strength. 'Badges' such as the size or colouring of a patch of feathers on the breast indicates their dominance. Weak fighters will have small badge and strong fighters will have a big badge. The question facing evolutionary behaviourists is, why don't the strong fighters just evolve big badges and pretend to be good fighters, scaring off the competition? In other words, why not bet big with a bad hand? Why not bluff?

Lachmann, Szamado & Bergstrom (2001) point out that these systems differ in a major respect to other agnostic signals: The property they signal about is easy to verify. If you doubt a competitor can really fight well, you can just step up and peck him in the face; you can 'call'. Other animal signals, such as the size of a deer's antlers, are good indicators because they are directly linked to the property they signal about - a deer with big antlers not only has better weaponry but must also be strong enough to support such a heavy structure. Bird signals are disconnected from their meaning - they are fragile. Lachmann et al. suggest that bird signals are kept honest because anyone can challenge them. A bird signaling beyond his station - 'bluffing' - will eventually be found out and risk injury or death from a fight it cannot win. Cheaters are punished, bluffers will be revealed and lose their chips.

However, I'd like to make another observation - the fact that competing birds and poker players don't know how their strength compares to others is important. It means that you can only asses your own strength by observing the behaviours of others. A pair of jacks might seem like a good hand, but if someone raises your bet, you'll reassess the situation. (Another way of saying this is that the signaller cannot observe its own signal - birds can't see the feathers on their own chest)

I'm wondering if honesty falls out of this kind of interaction. To test this, I ran a model. You could see it as a model of poker:
  • A group of people are playing poker. They each get dealt a hand. This represents their individual strength. However, they cannot see other people's hands. Each player guesses his relative standing.
  • Each player also has a 'bet' that he places with the hand. This represents the signal to competitors.
  • Players play against each other, and can choose to call or fold to another player's bet.
  • If a player calls another, they compare their cards and the player with the best hand wins.
  • The winner increases his estimation of his standing in comparison with the group, and the loser lowers his estimation. The poker analogy is eroding, but bear with me.
  • After several rounds, the players are replaced by onlookers who adopt the strategies of the most successful players, with a little variation. This represents evolutionary selection. Ok, so actually they inherit the player's cards as well, with a little variation. Yes, the poker analogy has gone bust.
This is a modified model of Winner and Loser Effects (Dugatkin & Druen , 2004; Dugatkin & Dugatkin, 2007).

Anyway, I predict that, over time, the player's bets will accurately reflect the player's hands. Good hands will place high bets and bad hands will place low bets. To switch back to signaling in birds, the size of the badge will correlate with the fighting ability of the bird.

Running this model reveals that, indeed, that is what happens. The graph below shows the correlation between badge size and fitness for 100 models with 50 individuals competing for 100 generations:

Point proven? Not so fast. Although the correlation between badge size and fighting ability does tend to increase, the standard deviations of badge sizes and fighting ability tend to decrease. That is, everybody becomes the same and there is no hierarchy. In this case, the signaling system has become redundant, and we can't say that honesty is stable. The problem lies in the amount of 'variation' with which players copy their 'parents' - that is, the genetic/ecological drift. Here's a graph of how the correlation varies as a function of the probability of a trait (e.g. player's hand or betting amount) changing and the amount it can change (each point is the mean of 10 runs on those settings):

We see that a high probability of a small amount of change leads to high correlations (stable honesty). However, the relationship is almost opposite for the standard deviation of badge size and fighting ability (each point in the graph below is the slope of a linear fit of the last 20 generations of 10 runs of the model with varying drift magnitude and probability, negative values indicate the hierarchy is eroding):

Combining the three graphs, we find a space where an stable honest, hierarchical system is possible. The magic setting is one with a high probability of a small amount of change:

Ok, so what does this mean? Firstly, it's possible that honesty is inherent in a system where individuals are blind to their actual relative strength and must use the interactions with others to establish their place in the hierarchy. Furthermore, variation is key to maintaining this system. Since this is just a blog and not a paper, I can make wild unjustified claims, so here goes: Another ponderously stable signalling system is human language. Despite the obvious lying that goes on, most utterances are essentially honest. Also, human languages are hugely varied. Furthermore, children appear to be designed to handle a large variation in the input - including several languages at once. Perhaps language is built upon the same foundations as signalling systems in birds. After all, language is cheap and arbitrary and learned by interacting and observing behaviour, just like bird 'badges'.

To apease the poker fans that have made it this far, let's go back to the table. The variation in poker comes from the cards on the table - they provide an extra unknown factor which introduces an element of chance into the game. What all the pretty graphs suggest is that without this element, players would cease to rely on betting as an indication of how likely the player is to win. In other words, the betting would become irrelavent and pointless, and the game wouldn't work.

So next time you get upset when someone's bluff lucks-out on the river, just remember - it's all part of the game.

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