Consider a Man and a Woman. Time flows continuously and the horizon is infinite. At time $T=0$ they are locked in an embrace, and every instant of time $t>0$ their lips draw closer. Let $\delta_t$ be the distance at time $t$, it declines monotonically over time.  At each $t$, the two simultaneously choose actions $a^i_t$ which jointly determine the speed at which they close the space that separates them, governed by the rule

$\frac{d \delta_t}{d t} = - f(a^M_t, a^W_t)$

where $f$ is strictly increasing in both arguments. In addition, both the Man and the Woman can pull away at any moment by choosing action $a_0$, thereby spurning the kiss and ending the game.

The closer they get the clearer they can see into one another’s eyes, revealing to each of them the true depth of their love, captured by the state of the world $\theta$ which they receive private, and increasingly precise signals about as the game unfolds.

In this game, the lovers have common interests. Each wants to kiss if and only if their love is true, i.e. $\theta >0$.  However, they know the risks of opening their heart to another:  neither wants to be the one left unrequited. When $\theta > 0$, each prefers kissing to breaking the embrace, but each prefers to pull away first if they expect the other to pull away.

Along the equilibrium path their lips move fleetingly close. At close proximity every tiny fluctuation in the speed of approach communicates to the other changes in the private estimates $\hat \theta_i$ each lover $i$ is updating continuously over time, i.e.  $a^i_t$ varies monotonically with the estimate $\hat \theta_i$.

But then: does he see doubt in her eyes? Did she blink? He cannot be sure. A bad signal, a discrete drop in his estimate and this causes him to hesitate.  And since $\theta$ is a common state of the world, his hesitation is informative for her and so she pauses too. Not just because his hesitation raises doubts that their love is everlasting, but worse:  he may be preparing to turn away.  She must prepare herself too.

But she doesn’t. She sees deeper than that and instead she lurches ahead ever so slightly. He is looking into her eyes:  he can see that she believes with all her heart that $\theta$ is positive. And now he knows that these are her true beliefs because if in truth her estimate of $\theta$ was close to the negative region, his hesitation would have pushed her over and she would have turned away pre-emptively. Instead her persistence implores him to have faith in their love and to stay there in her arms with his lips so tantalizingly close to hers.

His doubts are vanquished. He loves her. She knows that he knows that she loves him too. And at last it is common knowledge that their love is true and they will kiss and in their moment of deepest passion they discover something about their payoff functions they haven’t before. This moment is the first moment in the rest of their lives together. They will not rush. Time is standing still now. Together, as if coordinated by the eternal spirit of amor, they allow $a_t^i$ to fall gradually to zero, just slow enough that their lips finally meet, but just fast enough that, when they do,

$\frac{d \delta_t}{d t} \rightarrow 0$

so that their convergence occurs smoothly but still in finite time.

Happy Anniversary Jennie

(drawing:  Chemistry from www.f1me.net)