Play boldly when you're behind

Game theory involves a very specific setting:
players choose independently from a menu of actions with consequences depending on all choices in a known way.
This is a fascinating mathematical subject, and indeed the language of Prisoner's dilemma and Nash equilibrium has spread far beyond the mathematics world. There are many introductory textbooks and also many non-technical books such as Len Fisher's Rock, Paper, Scissors: Game Theory in Everyday Life. I don't have any general insights to add to such books. But notwithstanding the title above, there are in fact very few everyday life, or familiar game, settings which fit the specific setting of game theory. So instead of writing about game theory I will record a general principle which is relevant to the kind of games people actually play.
Play conservatively when ahead, play boldly when behind.
Of course this is much too well known to count as an insight. But one can readily use it to make testable predictions. For instance
In American football, classify interceptions by quarter and whether the offense was ahead or behind; of these 8 possibilities, the most frequent will be "4th quarter, behind".
And this is borne out by data.

Let me invent a numerical example where we can observe the principle, and where readers may test their intuition about just how boldly they should play. Players A and B have equal ability and take turns in which they try to score points. In each turn they have a choice of actions in attempting to score points, as in the table.

 number of points attempted 1 2 3 4 5 probability of success 0.5 0.22 0.13 0.08 0.05 Expectation of score 0.5 0.44 0.39 0.32 0.25

Intuition tells us to try for 1 point at the start (conservative play), and only change to trying for more points (risky play) if you are behind near the end of the game. To make the rules precise we say that a tie (equal points) is counted as a win for A, because B has an advantage in playing second. Games like this are very easy to analyze numerically via dynamic programming. The results are shown below, as optimal choice of action for B and B's probability of winning the game, depending on the number of turns remaining and the current point difference (B - A). The mathematics is easy because B's best choice on the final play is obvious, and then we work backwards to calculate the best choice for each player on each turn. The numbers seem roughly in accord with intuition.

(optional action ; chance of winning) for B
Point difference Turns remaining
5 4 3 2 1
+3 1 ; 0.88 1; 0.89 1; 0.91 1; 0.91 1; 1.0
+2 1 ; 0.81 2; 0.83 1; 0.86 1; 0.82 1; 1.0
+1 1 ; 0.71 1; 0.73 1; 0.76 1; 0.55 1; 1.0
0 1 ; 0.55 1; 0.55 1; 0.56 1; 0.30 1; 0.50
-1 1 ; 0.38 1; 0.37 1; 0.34 2; 0.19 2; 0.22
-2 2 ; 0.25 3; 0.24 3; 0.23 3; 0.12 3; 0.13
-3 3 ; 0.16 3; 0.15 4; 0.14 4; 0.06 4; 0.08
-4 4 ; 0.10 4; 0.08 5; 0.07 5; 0.02 5; 0.05