FIFA experimented with a “sudden-death” overtime format during the 1998 and 2002 World Cup tournaments, but the so-called golden goal was abandoned as of 2006.  The old format is again in use in the current World Cup, in which a tie after the first 90 minutes is followed by an entire 30 minutes of extra time.

One of the cited reasons for reverting to the old system was that the golden goal made teams conservative. They were presumed to fear that attacking play would leave them exposed to a fatal counterattack.  But this analysis is questionable.  Without the golden goal attacking play also leaves a team exposed to the possibility of a nearly-insurmountable 1 goal deficit.  So the cost of attacking is nearly the same, and without the golden goal the benefit of attacking is obviously reduced.

Here is where some simple modeling can shed some light.  Suppose that we divide extra time into two periods.  Our team can either play cautiously or attack.  In the last period, if the game is tied, our team will win with probability p and lose with probability q, and with the remaining probability, the match will remain tied and go to penalties.  Let’s suppose that a penalty shootout is equivalent to a fair coin toss.

Then, assigning a value of 1 for a win and -1 for a loss, p-q is our team’s expected payoff if the game is tied going into the second period of extra time.

Now we are in the first period of extra time.  Here’s how we will model the tradeoff between attacking and playing cautiously.  If we attack, we increase by G the probability that we score a goal.  But we have to take risks to attack and so we also we increase by L the probability that they score a goal.  (To keep things simple we will assume that at most one goal will be scored in the first period of extra time.)

If we don’t attack there is some probability of a goal scored, and some probability of a scoreless first period.  So what we are really doing by attacking is taking an G-sized chunk of the probability of a scoreless first period and turning it into a one-goal advantage, and also a L-sized chunk and turning that into a one-goal deficit.  We can analyze the relative benefits of doing so in the golden goal system versus the current system.

In the golden goal system, the event of a scoreless first period leads to value p-q as we analyzed at the beginning.  Since a goal in the first period ends the game immediately, the gain from attacking is

G - L + (1-G-L)(p-q).

(A chunk of sized G-L of the probability of a scoreless first period is now decisive, and the remaining chunk will still be scoreless and decided in the second period.)  So, we will attack if

p - q \leq G - L + (1 - G - L) (p-q)

This inequality is comparing the value of the event of a scoreless first period p-q versus the value of taking a chunk of that probability and re-allocating it by attacking.  (Playing cautiously doesn’t guarantee a scoreless first period, but we have already netted out the payoff from the decisive first-period outcomes because we are focusing on the net changes G and L to the scoring probability due to attacking.)

Rearranging, we attack if

p - q \leq \frac{G-L}{G+L}.

Now, if we switch to the current system, a goal in the first period is not decisive.  Let’s write y for the probability that a team with a one-goal advantage holds onto that lead in the second period and wins.  With the remaining probability, the other team scores the tying goal and sends the match to penalties.

Now the comparison is changed because attacking only alters probability-chunks of sized yG and yL.  We attack if

p - q \leq Gy - Ly + (1 - G - L) (p-q),

which re-arranges to

p - q \leq y\frac{G-L}{G+L}

and since y < 1, the right-hand side is now smaller.  The upshot is that the set of parameter values (p,q,y,G,L) under which we prefer to attack under the current system is a strictly smaller subset of those that would lead us to attack under the golden goal system.

The golden goal encourages attacking play.  The intuition coming from the formulas is the following.  If p > q, then our team has the advantage in a second period of extra time.  In order for us to be willing to jeopardize some of that advantage by taking risks in the first period, we must win a sufficiently large mass of the newly-created first-period scoring outcomes.  The current system allows some of those outcomes (a fraction 1-y of them) to be undone by a second-period equalizer, and so the current system mutes the benefits of attacking.

And if p<q, then we are the weaker team in extra time and so we want to attack in either case.  (This is assuming G > L.  If G< L then the result is the same but the intuition is a little different.)

I haven’t checked it but I would guess that the conclusion is the same for any number of “periods” of extra time (so that we can think of a period as just representing a short interval of time.)