Dissertation · one idea, six settings

The Life of a Spike

A player does something sudden and unexpected, once. The others notice it only through noise, and respond. Most of what separates the chapters is what happens to that spike afterwards.

age of the spike 0.00 when in the game (Chapters 1–2)

Drag the age slider, or drag across any chart. Milestones light up as the spike passes them; one that the game ends before is struck out.

Each row is the response to a single spike. The player who made it knows it did, and carries on playing its best given that (Chapter 6’s blip convention); everyone else keeps their equilibrium plan and reacts to what they see. The finite games last 1 time unit, so a spike’s life there depends on how much of the game is left, and “when in the game” moves it. The stationary games have no clock, so every spike lives the same life. Each row is one small example of its chapter’s model, described under it, with parameters chosen to make the picture clear rather than the chapter’s own. Computed with noisestate (its development version); each row’s vertical scale is its own.

Two ways to carry on, one equilibrium

After a spike, the player who made it can hold still (a frozen spike) or keep playing its best knowing what it did (a blip). The lives above use blips. The two tell different stories, yet they pick out the same equilibrium, because each can be built out of the other: they are two ways of listing the same moves. Here is that, in the numbers of Chapter 3’s game.

frozen spikes added 0

size of the change

So the first-order test is the same in either basis. Player 1’s extra cost from a small change to its strategy, then holding still or playing on: both curves are flat at zero, because a move that helped in one list would be a combination of moves in the other, and none of those helps. The two part only at second order: holding still after a change costs more. Counted as if nobody reacted, the same change looks profitable; the bottom is flat only because the others react.

When someone is watching

With nobody privy to the slip, how the deviator carries on after it is invisible to the others: they see only its effects, and the two conventions describe the same equilibrium. A privy player sees the slip itself, so its best reply depends on what it expects the deviator to do next. Holding still would itself be a string of further slips, each of which the privy player sees and answers; playing on is not. Chapter 6 settles this with the blip: privy players expect the deviator to treat the slip as bygone and carry on playing its best, not to read it as a sign of what the deviator will do later. The deviator’s own first-order condition is the same under either convention; what the privy players expect, and so how they respond, is not. That is why the lives above use the blip, and why Chapter 6’s privy trader sells into the bad quote expecting the market maker to go on working off its inventory.

Chapter 6’s privy trader after the market maker’s bad quote, in the same equilibrium, under the two expectations. Both begin with the same block of 2.5. Expecting the market maker to play on, the trader buys a little back as the quote comes down. Expecting it to hold the bad quote, the trader keeps selling into it, harder and harder, and the inventory runs away.

The noise-state, explained The information wedge The dissertation