A doctor tells you that you have six months to live. Six months where you feel perfectly fine. Maybe you quit an unfulfilling job, see the world, spend more time with friends and family.
The strangers you pass don’t know. For them the world carries on as usual, so your new plans are made for a world where everyone else acts mostly the same.
Now everyone finds out, credibly, that in six months the sun will explode. It makes a world of difference. If you planned to tour the world, who is going to fly the plane?
Everyone is making the same kind of plans you are, and it would be foolish to make yours assuming the world stays the same. Pandemonium.
Now only five researchers find out. They look through a telescope, do the calculations, and reach the same conclusion. The news affects everyone, and nobody else knows. If they all quit their jobs and travel the world, it will be obvious something happened, and then pandemonium.
So they act quietly. What they do is how the rest find out.
Put that into the textbook model. Many people push on one state, and each acts on their own estimate of it, the way a single controller would.
Now the state moves with everyone’s estimates, so to choose an action you have to anticipate what the others estimate.
Your actions depend on what you believe about the world, and on what you think others believe. If everyone acts on their beliefs about others’ beliefs, those become relevant too: beliefs about beliefs about beliefs. There is no last layer.
With n players there are n beliefs of the second order, n² of the third, and so on. Townsend called it forecasting the forecasts of others.
Change what a belief is a belief about. Underneath everyone’s model of everyone else are the same sources of randomness. Estimate those, and the nested models are no longer needed.
In a linear-quadratic-Gaussian game nothing moves on its own. Every price, belief and action moves because some primitive shock hit it: a surprise in demand, a trader’s news, the noise in a signal.
Call the shocks W. Above them, the state X they push around.
Hit the system with a single shock at time s and watch what follows. The trace it leaves is the impulse response L(t, s): how much of a shock at s still shows at t.
For an Ornstein–Uhlenbeck process it is e−(t−s); here, its discrete cousin 0.92t−s.
Linearity means responses add. Any quantity is its mean plus the responses to every shock so far, each scaled by the size of its shock.
Kick by kick, one kernel rebuilds the whole path.
You can always make a process Markov by carrying its whole past, but which past is enough differs for every process. The shocks are underneath all of them, so track W. A player never sees it; from a noisy signal they estimate its whole path up to now, and each new observation revises old estimates too.
That estimate, a belief about the shocks rather than about the state, is the noise-state. The shocks already happened and do not move, so updating it needs no prediction step, only the news.
To forecast anything, a player runs the same kernel against their estimated shocks instead of the true ones. Conditioning changes the integrator and nothing else.
Now the tower. Player 2’s forecast is itself a kernel against the shocks, so player 1’s forecast of it is that kernel run against player 1’s noise-state. A forecast of a forecast is a composition of kernels; on a grid of dates, a matrix product.
The regress closes, at any depth, without cutting the loop.
Strategies are kernels too, and the best response to kernels is a kernel. Equilibrium is a fixed point in kernels, found by iterating until they stop moving.
Those kernels are the impulse responses the explorer plots, and what the chapters compute.