More on Moore’s Law

In this blog post, we study the conditions for which a signal can be trusted when sent to a distrustful party.

The behavior of animals in the wild is often puzzling. Why do babies cry so loud? Why do gazelles jump vertically when they see a predator? It turns out these are primitive forms of communication in the form of signals. Gazelles signal their health and fitness, babies signal their hunger or fear.

But… How do these signals come about?

A branch of mathematics called Game theory provides an insightful framework for understanding how these behaviors come to be, and why they are the way they are. Seen through the lens of Game theory, the previous examples are forms of signaling games.

A signaling game, as defined in Wikipedia, is a dynamic game in which two players, the sender (S) and the receiver (R), interact. The sender has a certain type τ, which is given by nature. The sender knows his own type while the receiver does not know the type of the sender. Based on his knowledge of his own type, the sender chooses to send a message from a set of possible messages M = {m₁, m₂, m₃,…, m_j}. The receiver observes the message but not the type of the sender. Then the receiver chooses an action from a set of feasible actions A = {a₁, a₂, a₃,…., a_k}. The two players receive payoffs dependent on the sender’s type, the message chosen by the sender and the action chosen by the receiver.

An example from nature

Lets consider a lion chasing a gazelle. The observer would notice that in such cases, the gazelle, upon detecting the lion, will start stotting. By doing so, it is signaling its fitness and probable ability to outrun the lion. The lion can then decide not to chase the gazelle, and wait for another (better) opportunity.

Both the lion and the gazelle have an interest in avoiding unsuccessful chases. Both lose energy during the chase, and the gazelle loses out doubly as the time spent running is time not spent grazing. Therefore evolution puts pressure on these species to develop something in order to avoid this particular outcome. This something is a signal-producing capability, that provides orchestration by communication.

Framing the game

In this example, the signal sender S is the gazelle, and the signal receiver R is the lion. The signal is either stotting m₁ or no stotting m₀, so M = {m₀, m₁}. The lion’s feasible actions are chase a₁ or ignore a₀, so A = {a₁, a₂}. The payoffs are chance to catch for the lion, and chance to get caught for the gazelle, which are a function of the gazelle’s fitness τ, the action taken by the lion, and the signal emitted m_i; hence P_lion= f(τ, a_j, m_i), and P_gazelle= g(τ, a_j, m_i). We can now draw the payoff matrix:

figure 1: payoff matrix in a signaling game

Where m₀= 0, and m₁= m.

Conditions for honest signaling

Now that we have framed the example, we can analyze the different strategies available to the players. Three factors influence the payoff outcome as we have seen; the gazelle’s fitness, the signal it chooses to send (as defined by the type of game, and since there would be no use in sending it if it had no impact), and the lion’s reaction.

Lets examine the conditions to which signaling is beneficial to both parties. That is, how does signaling drive out wasteful unsuccessful chases, which use up energy for nothing, through evolutionary pressure?

For this, we must establish relationships between fitness, energy expenditure due to chasing or fleeing, and emitting the signal. We’ll function on the following relationships for the fitness cost of the signal C_signal and the fitness cost of a chase C_chase:

C_signal = h(τ, m) and C_chase = k(τ)

If the signal m_i has an effect on fitness that is not related to τ, then

C_signal = h(τ, m) = h(τ – m)

and

P_gazelle= g(τ, a_j, m_i) = g(h(τ, m_i), a_j)  = g(τ – m_i, a_j)

If we furthermore assume that a₀has no effect on payoff, then

g(τ, a₀) = g(τ) and g(τ – m, a₁) = g(τ – m)

We can then establish the following payoff matrix:

figure 2: gazelle payoff matrix if signal has an invariant effect on payoff

Lion payoff is calculated with function f instead of g.

We can now calculate the difference in fitness between with and without the signal for the lion and the gazelle. Lets assume that f and g are linear functions.

If lion does not chase, then

Δ_fitness= g(τ) – g(τ – m) = g(m)

but if it does, then

Δ_fitness= g(τ  – k(τ)) – g(τ – k(τ) – m) = g(m)

The lion will chase the gazelle if its payoff for chasing is larger than payoff for not chasing

f(τ  – k(τ)) > f(τ)

f(τ  – k(τ)) – f(τ) > 0

f(k(τ)) > 0

The gazelle will emit the signal if its payoff for emitting the signal is larger than for not emitting,

In the case the lion chooses to chase, then

g(τ  – k(τ) – m) > g(τ)

g(τ  – k(τ)) – g(τ) > 0

g(k(τ)) > 0

In the case the lion chooses not to chase, the calculation is very much the same.

As we can see, it does not make sense for the gazelle to send a message in this particular case. This is because we are analyzing a single occurrence of a signaling game. In reality, the lion and gazelle face a repeat game. For example, Tit for Tat is not a strategy for single shot prisoner’s dilemma games, but it is optimal for repeat ones.

What do we learn from this?

That Game theory provides us with a framework for understanding the theoretical underpinning of phenomena observed in nature, and helps build models of prediction for anticipation and comprehension.