Before the Opening Whistle: How Mathematics Shape the Incentives in Group-Stage Matches in the Reformed FIFA World Cup

Two researchers from HUN-REN Institute for Computer Science and Control, Senior Research Fellow László Csató and Research Fellow András Gyimesi, have examined what to expect in the last round of the group stage using a novel model. László Csató talked about the research background, famous anomalies, and the surprising findings of their simulations.

– When fans talk about football, operational research models or Poisson distributions rarely come up. How does mathematics find its way into a sport as unpredictable as football?

– It may seem surprising at first, but an enormous amount of strategic and mathematical reasoning underlies the rulebook of modern sport. One of the most important questions in sports-related academic research is whether players or teams face the appropriate incentives – whether the rules genuinely require them to strive for victory in every match, or whether they open the door to tactical manoeuvring or even to deliberate losing. The World Cup group stage –in particular, its final round – is especially interesting for researchers in this regard, because the mathematics of qualification frequently overrides straightforward sporting logic.

– Why does the research focus specifically on the last round of the group stage?

– Because that is when incentives become the most clear. Our research proceeds from a fundamental, empirically well-supported assumption: the primary objective of a team is to secure qualification from the group. Once that is achieved, it matters relatively little in which position they advance – winning the group is not always beneficial in the knockout stage, since everything depends on the bracket and the opponents. If qualification is the only objective, the final round can give rise, in the simultaneously played matches, to situations where the interests of the teams diverge sharply from what one would ordinarily expect.

The most famous – and most scandalous – example is the West Germany–Austria match at the 1982 World Cup in Spain, called since the "Disgrace of Gijón". Going into that final group match, the standing revealed that a West German win by one or two goals would see both sides advance at the expense of Algeria. What happened on the pitch? Germany scored in the 10th minute to lead 1–0. For the remaining 80 minutes, essentially nothing happened; the teams passed the ball about and strolled around the pitch. The spectacle was so indefensible that a television commentator openly urged viewers to switch off, while one German supporter publicly burnt his national flag. That notorious incident led to the decision to play the final two group matches simultaneously.

– How does your proposed new probabilistic model work? How were you able to classify these complex situations into categories?

– The previous literature followed a binary, deterministic approach: a match was either stakeless for a team – because they had already qualified or were eliminated –, or their prize remained uncertain. We propose a more sophisticated probabilistic model based on a cost–benefit analysis. The model assumes that, in the final round, teams know the results of earlier matches, and the 0–0 scoreline serves as their reference point. From that position, we ask: if a team changes tactics and plays more offensively, how much does the increased chance of scoring raise its probability of qualification – and, conversely, how much does it risk, since a more attacking tactic also means that conceding some goals becomes more likely, potentially leading to defeat and elimination.

– On that basis, what types of match can you distinguish? The number of possible outcomes is clearly finite.

– Each of the two teams on the pitch has one of three possible cases of incentives: it prefers attacking; it prefers defending; or it is essentially indifferent to the result, because qualification or elimination is already guaranteed with a very high probability. Combining these for the two opponents yields six distinct match types.

The first is what we call an unimportant match – one where the outcome is irrelevant to the qualification prospects of either side. Then there are two asymmetric cases: defensive asymmetric, where one team is indifferent and the other needs to protect the result of 0-0 – that is, to defend; and offensive asymmetric, where the opponent of the indifferent team prefers attacking.

In the remaining three categories, both teams still have something at stake. In an offensive match, both sides need to attack – these are perhaps the most enjoyable for supporters, since they can be expected to produce open, high-tempo, end-to-end football. Next is the antagonistic match, where a draw suffices for one team – so that it will focus on defence – while the other must win and is therefore compelled to attack. Finally, the most threatening is the defensive match: both teams are relatively satisfied with a draw of 0-0 for qualification purposes, so neither is willing to take risks. Dull football is thus effectively baked in, and it is here that the risk of tacit collusion, such as in the case of Disgrace of Gijón, becomes the greatest.

The practical relevance of our model is demonstrated by the fact that, according to our historical simulations, all six match types actually occurred at the 16 matches played in the last round of the 2018 FIFA World Cup group stage. These are not merely theoretical curiosities – they reflect how football genuinely behaves.

Take an example from the 2022 World Cup. France had already secured qualification after two matches; their final-round opponent, Tunisia, could still have reached second place with a win, provided Australia did not beat Denmark. In our model, this is an offensive asymmetric match: stakeless for France, while Tunisia needs to win to have any chance of progressing. The French coach could accordingly afford to field nine new players, resting his key men – and, as we know, France duly lost by 1–0.

– Let us turn to the reform that prompted your research: FIFA is making fundamental changes to the 2026 tournament. What are the main modifications, and how did you decompose their effects?

– The reform is a complex package involving three significant changes, which take effect simultaneously but independently of one another. A major novelty of our research is that, for the first time in the academic literature, we were able to quantify and separately evaluate the effect of each of these three factors.

The first change is the expansion: from 32 to 48 teams, divided into 12 groups of four. The second is the modified qualification rule: in addition to the group winners and runners-up, the eight best third-placed teams also advance to the knockout stage, the Round of 32. The third is the changing of the tiebreaking rules: goal difference was previously the primary criterion to separate teams with an equal number of points, but it has now been replaced by head-to-head results.

– Starting with the expansion to 48 teams: what do the simulations show about its impact on the final round of the group stage?

– The effects of the expansion are, to put it mildly, worrying for the quality of play. By bringing in 16 weaker teams, the strength differences within the groups increase substantially and competitive balance worsens. Our simulations show that the expansion alone shifts 10–12% of all matches from the exciting antagonistic category into the wholly unimportant one (see Figure 1). Why? Because the greater disparity in quality implies that it becomes more likely for the two stronger teams to secure qualification after just two rounds, while the two weaker ones are already eliminated – leaving the final-round matches essentially without stakes. But this is purely the effect of expansion; it is not the full picture.

– What about the second change – the possibility of third-placed teams qualifying?

– This is the most positive development, because it fundamentally rewrites the mathematics. Based on our simulations and historical data, a single win – three points – already provides a high chance of advancing from the group. Hence, almost no team will be eliminated before the final round. This rule change substantially reduces the number of antagonistic matches and markedly increases the proportion of offensive matches, in which both sides must play for a win. This is excellent news for supporters, as it is likely to produce far more high-tempo, tension-filled encounters.

– What about the third change? FIFA is making head-to-head results the primary tiebreaking criterion in place of goal difference. Why might they have decided this, and does the science suggest it will pay off?

– FIFA presumably wanted to give greater weight to direct clashes between rivals, making teams even more motivated to win those encounters. Our simulations, however, unambiguously show that using head-to-head results is considerably worse than using goal difference.

According to our model, this seemingly minor rule change increases the frequency of unimportant matches in the final round by more than 5 percentage points – a far from negligible rise, and a clearly negative development. Head-to-head results effectively freeze positions early: if team A beats team B in the first round, then in certain equal-points scenarios it is mathematically impossible for B to overtake A, regardless of how many goals it scores in the final round. This kills the excitement and reduces the range of possible end-of-group outcomes.

– If we combine the three effects, what is the aggregate verdict? Will the final round be better or worse overall?

– The picture is mixed. Looking at Figure 2, the most striking change is that the probability of unimportant matches is effectively doubled compared to previous World Cups, rising from below 20% to almost 35%. This is a robust increase: of the 24 simultaneously played final-round matches, roughly eight are expected to be irrelevant with respect to qualification. That is an unambiguously negative development.

On the other hand, the structure of competitive matches also changes: the proportion of harder-to-digest, tactical antagonistic matches is halved – falling from around 40% to below 20% –, while the proportion of entertaining, pure offensive matches roughly triples, rising to between 15 and 20%.

– The integrity of sports is always a concern for fans and organisers. How does the risk of match-fixing – or Gijón-style tacit collusion – change under the new system?

– This is also a key question. The good news is that in offensive and antagonistic matches – where at least one team must attack, and the other also has something at stake – any manipulation is virtually out of the question. The risky zone is defensive matches, where the result 0–0 is ideal both sides; although their frequency shows some increase, their share remains low, below 5%.

As for the doubled number of unimportant matches, there is actually less incentive for fixing or collusion, precisely because the outcome does not count for the two opponents. The greater risk there is simply that the lack of stakes discourages full effort, reducing the quality of play. All in all, the risk of deliberate wrongdoing has not increased dramatically – however, the number of lukewarm, unengaging matches unfortunately has.

– Is there any factor that could further refine these results, but is outside the scope of the current research?

– Yes: the match schedule, the ordering of fixtures within the group. It makes a considerable difference whether the two strongest teams meet in the first round or whether they play against each other only in the last, third round. This variable – currently determined by a random draw – can fundamentally affect the probabilities we have calculated. We deliberately excluded it from the present paper, but it provides a promising direction for future research: can the optimal schedule reduce the number of stakeless or defensive matches in the final round?

– In the simulations, you computed probabilities not only for the current matches, but also did not take actual results for the previous World Cups. Why was this necessary?

– Because in football, reality is only one single realisation of many random variables. Take the 2022 FIFA World Cup: the eventual champion, Argentina, lost their opening match to Saudi Arabia. It would be a serious methodological error to consider this result as given – if that match were replayed a thousand times, Argentina would win in more than 80% of the simulations. With relying solely on what actually happened – that are often outliers – organisers and researchers would risk building entirely misguided rules. That is why we simulated every historical World Cup, as well as the 2026 format, using an Elo rating-based Poisson model, and running each simulation a thousand times, to obtain a representative distribution of incentives.

– Finally, what is your message to international governing bodies such as FIFA or UEFA?

– The most important lesson is that rule changes always have hidden mathematical and strategic consequences that must not be overlooked. In a similar complex reform package, well-intentioned decisions – such as increasing the number of participating teams or prioritising head-to-head results – can prove counterproductive and cancel out the positive effects of other rule changes. We encourage governing bodies in sports to carry out in-depth simulation analyses before future rule changes are implemented, or to strengthen the collaboration with the academic community. Decisions that will shape the future of the sport should be taken with the findings of scientific research factored in, with a careful consideration of advantages and disadvantages.

The paper underpinning this interview is available here.