Trust, Signals, and the Evolution of Cooperation

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«For [philosophers] conceive of men, not as they are, but as they themselves would have them be. Whence it has come to pass that, instead of ethics, they have generally written satire …»

Baruch Spinoza, „Tractatus Politicus“

What would be a theory of the social contract that treats men as they are? Not one that treats the social contract as a single, all-explaining principle. Rather, it should see a social contract as an interacting web of conventions and norms. Norms are conventions backed by sanctions. Sometimes norms are written in the law and enforced by the government. Sometimes they are implicit, and the sanctions take the form of social pressure.

Norms as Evolving Conventions

Conventions, and thus norms, evolve. This «evolution» includes cultural evolution. It is then useful to explore issues of the social contract using tools of evolutionary game theory. Game theory studies how people make decisions that depend on others. It arose in the work of von Neumann and Morgenstern as a theory of economic behavior based on idealized high rationality. Evolutionary game theory started in biology as a theory of how strategies spread over time when success depends on interaction with others. It now contains all sorts of adaptive dynamics as is appropriate for its extension to cultural evolution. Understanding how social contracts form can also help us understand how they sometimes fall apart.

Evolutionary Game Theory as a Framework for Social Order

Evolutionary game theorists start with simple strategic interactions, modeled as simple games that evolve by simple adaptive dynamics. These models can then be used as building blocks for more complex models of interactions and of interaction networks. Evolutionary game theory is a large, growing field. Here are a few examples.

Consider what game theorists call the Stag Hunt game. The game is adapted from a story in Rousseau’s Discourse on Inequality. In it, one can either hunt hare or hunt stag. Hunting hare can be done alone, does not require coordination, and one’s chances of success are the same no matter what others do. Hunting stag alone always fails. It requires coordinating with a team for success. But a successful Stag Hunt carries a larger reward per hunter.

A 2-person Stag Hunt game would look like this: Hunting stag with another stag hunter gets a payoff of 4. Hunting stag alone yields 0. Hunting hare always gives a payoff of 3, regardless of the other player. Stag Hunt games with more people are also possible, but we can illustrate some general points with this simple 2-person example.

There are two strict equilibria in this game. An equilibrium is a situation where no player can benefit by changing their choice alone. If both players hunt hare, it is an equilibrium because if the other hunts hare, it is best for you to hunt hare too. If you switched to hunting stag, you would go from a payoff of 3 to 0. Both players hunting stag is also an equilibrium because if the other hunts stag, it is best for you to hunt stag. If you switched to hunting hare, you would go from a payoff of 4 to 3. Hunting stag creates the best expected payoff for everyone. But it also carries the risk that the other player will go off and hunt hare instead, leaving you with nothing. Hunting hare, on the other hand, carries no risk. This creates a trade-off between higher rewards and risk–a question of trust. If both players trust each other to cooperate, then they will cooperate.

Suppose that we have a very large population of some stag hunters and hare hunters with random encounters between them. Over time, the population may settle into either equilibrium, depending on where it starts. If there are a lot of hare hunters, hare hunting will take over. If there are a lot of stag hunters, stag hunting will take over. With the example given, a somewhat greater number of initial hare hunters lead to the less risky hare-hunting equilibrium. If we switch our attention to a finite population with some «mutation» or exploration, the population will settle near one or the other equilibrium for a while but will eventually bounce away from it and settle near the other. Cooperation will wax and wane. If the exploration is rare, then the periods near an equilibrium will be long, as it takes longer for the information of how easy it is to collaborate to spread. But with the example given, where hunting hare is «risk dominant», in the long run most of the time is spent in the hare-hunting equilibrium.

From Random Encounters to Social Networks

Everything is changed when encounters are not random. This is a crucial point in evolutionary game theory where it can depart radically from rational-choice based game theory. One way in which interactions may depart from randomness is in a setting where individuals learn with whom to interact. Consider a small finite group in which individuals start out meeting at random but adjust who they interact with through reinforcement learning. Then stag hunters rapidly learn only to interact with other stag hunters. Hare hunters don’t mind what others do, but since stag hunters avoid them, they settle into interacting with other hare hunters. In this way, networks of social interaction can form and tip the scales in favor of stag hunting.

Now let us add a strategy revision to this network. Individuals sometimes observe others in the network, see which strategy is doing better, and copy it. If the network is fast and the strategy revision is slow, the whole population can be eventually converted to stag hunting. Stag hunters rapidly find each other and prosper. Hare hunters are left to themselves and do less well. Slowly, by imitation, hare hunters convert to stag hunting. Everyone ends up hunting stag. But if the network is slow, the picture is not so rosy. The network can stay near random encounters for a long time with hare hunters doing better by playing it safe. Stag hunters convert to hare hunting. This is so because stag hunters can’t find each other often enough.

Signals and the Emergence of Meaning

The previous paragraph illustrated the interesting results of combining the Stag Hunt game with dynamic networks. Next, we will look at combining the Stag Hunt with signaling. But first, a quick introduction to signaling by itself. Signaling games of the kind we discuss were introduced by the philosopher David Lewis in his book, Convention. Here is the simplest stylized signaling game: Nature chooses one of two states with equal probability. One player, the sender, observes the situation and dependent on the situation observed sends one of two signals to a second player, the receiver. The receiver guesses the situation based on the signal and acts accordingly. The players have a common interest. If the receiver guesses correctly, they both get a payoff of 1, otherwise 0. There is no noise or cost to communication. Players have a problem different from those considered by Claude Shannon when he created information theory. The signals have no pre-existing meaning; the meaning emerges as players interact.

As before, there are multiple strict equilibria. There are two signaling equilibria, in which the players always get it right. In one equilibrium, signal A means state 1 and B means state 2. In the other, B means state 1 and A means state 2. There are also equilibria in which no information at all is transferred. The meaning of a signal, when it can be said to have one, depends on which signaling equilibrium the players are in. Meaning is conventional. Lewis discusses how the signaling can be sustained by high rationality. In contrast, we would like to know if they can originate from low rationality reinforcement learning.

Behavioral economics experiments of the laboratory show that humans learn quickly to signal in this game. Computer simulations of players using trial-and-error reinforcement learning quickly approach a signaling equilibrium. In fact, it has been proved that reinforcement learning always leads to perfect signaling in this simple game. Meaning arises spontaneously.

There is an interesting theory of more complicated signaling games. In it, multiple signalers interact in networks to process information and to coordinate teams. Here we just take the simple step of allowing our players in the Stag Hunt game to send signals and change their act based on the signals received. We will not equip the signals with any pre-existing meaning. As before, the meaning will have to arise spontaneously. An individual’s strategy now specifies what signal to send, what to hunt if you receive signal A or B. Individuals meet at random, they exchange signals, and then they choose to hunt stag or hare. Payoffs are then determined by the Stag Hunt game.

In a large population with random encounters, where you end up depends on where you start, but the proportion of initial states that lead to everyone hunting stag is dramatically increased – what we call a basin of attraction. This is because one type of hunter can now use a signal to hunt stag with like-minded hunters. These types outcompete others and can take over the population. They hunt stag with each other, so everyone hunts stag.

There is an apparent problem. There are two ways that this «handshake» could be done, depending on which of the two signals is chosen to mean «I will hunt stag with you if you are like me.» The answer to this difficulty is that which «handshake» wins out depends on initial population proportions. There will almost always be some initial imbalance, so that one convention will have a head start and will win out. Meanings still arise spontaneously.

In the finite population setting, the conclusion of the large population model survives. Signaling favors stag hunting. An unexpected bonus is that in this setting pre-play signaling also works for the infamous Prisoner’s Dilemma game. In this game there is only one equilibrium. It is always best for you to defect, no matter what the other does. This is so even though everyone is better off if everyone cooperates. For a numerical example, suppose that cooperating with a cooperator gets a payoff of 3, defecting with a defector gets a payoff of 2, but when a defector interacts with a cooperator the defector gets 4 and the cooperator gets 1.

The reason that evolution can nevertheless sustain some cooperation is again the handshake. Cooperators can now use a handshake to cooperate with cooperators and defect against defectors. There is a difficulty here that is not present with the Stag Hunt game. Defectors who «fake» being cooperators do even better than the natives in a population of cooperators using the signal. They can take over. This has been thought by some to be fatal for cooperation in this setting, but that is not quite correct. If the defectors who co-opt the signal take over, then cooperators who use a different signal as a handshake will start growing. It is an unending race. Analysis shows that some cooperation can persist. The more signals are available, the better for cooperation.

These examples give a taste of how evolutionary game theory can be employed. There are other simple games that are of interest in modeling aspects of the social contract: bargaining, partnership, division of labor, resource competition, exhibiting ownership. These can all be combined with each other, with signaling, and with social networks, both static and dynamic. Games themselves can evolve. The invention of new signals, for example, can have a powerful effect in complex signaling games, effecting escape from suboptimal equilibria. Invention of new acts can lead from all-or-nothing resource competition to bargaining. A wide range of adaptive dynamics remains to be fully explored. All of these pieces can form parts of a naturalistic theory of social contracts. A social contract is not a rational agreement imposed from above, but an evolving system of norms and conventions that emerges from repeated interactions, influenced by trust and adaptive dynamics.