Abstract: Penalty shootouts often decide the outcome of
important soccer matches. Although usually referred to as ”lotteries”,
there is evidence that some national teams and clubs consistently
perform better than others. The outcomes are therefore not explained
just by mere luck, and therefore there are ways to improve the average
performance of players, naturally at the expense of some sort of
effort. In this article we study the payoff of player performance
improvements in terms of the performance of the team as a whole.
To do so we develop an analytical model with static individual
performances, as well as Monte Carlo models that take into account
the known influence of partial score and round number on individual
performances. We find that within a range of usual values, the team
performance improves above 70% faster than individual performances
do. Using these models, we also estimate that the new ABBA penalty
shootout ordering under test reduces almost all the known bias in
favor of the first-shooting team under the current ABAB system.