Picking the joint over the marginals: a conjunction prediction engine for the 2026 World Cup

The Deel Kick Off contest attached to the 2026 World Cup pays a shared pool to entrants who correctly predict four outcomes at once: the champion, the Player of the Tournament, the top scorer, and that scorer’s exact goal count. An entry wins only if all four answers come true together, so the task is to find the likeliest combination of answers rather than the favourite for each question. We describe an engine that simulates the full 48-team tournament 50,000 times and picks the complete four-part ticket that comes up most often. On the champion, the one question with a betting market to test against, the engine beat the devigged closing line in three of the four World Cups from 2010 to 2022. Its mean log-loss was 1.897, against the market’s 2.199. The hardest of the four questions is who finishes top scorer, where the favourite holds only 24 percent of the probability and the race is as open as a field of about eleven equally likely players. The goal count is better located: on average the model puts 62 percent of its probability on five to eight goals, the range every Golden Boot from 2010 to 2022 fell in. The 2026 ticket the engine picks has a 0.94 percent chance of winning, roughly ten to twenty times a casual entry’s and still small in absolute terms. The main limitation is the format: no World Cup has been played with 48 teams, so the bracket itself cannot be backtested, and the engine never locks a real contest entry.

A free-entry contest run alongside the 2026 World Cup, the “Deel Kick Off” pool, is reported to award up to 150,000 dollars. To win a share, an entrant must correctly name four things before the tournament begins: the champion, the Player of the Tournament, the Golden Boot winner, and the exact number of goals that top scorer finishes on. The pool is split among every entrant who gets all four right. A single operator holds one ticket, with no second entry to hedge.

The structure changes the problem. A leaderboard rewards being roughly right about many separate questions, while this contest pays only for being exactly right about one combined event, all four answers coming true together. The obvious entry picks the favourite for each question on its own. That entry is usually not the likeliest combination, because the questions are linked: a striker whose team goes deep plays more matches and has more chances to finish top scorer.

The engine instead picks the likeliest complete ticket from a simulation of the whole tournament (Eq. 2), and one external test supports it: its champion forecasts beat the devigged betting market on past World Cups. We describe the simulation, the calibration evidence behind the joint estimate, and the size of the resulting probability.

Section 3 places the engine against the football-forecasting and odds-devigging literature, and Section 4 specifies the simulation, the team-strength model, the player models and the scoring rules. Section 5 reports the calibration results and the selected 2026 ticket, and Sections 6 to 8 interpret the findings, set out what the work does not establish, and say where the next iteration should go.

Statistical models of football scores begin with Maher (1982), who modelled goals as independent Poisson counts driven by team attack and defence parameters. Dixon & Coles (1997) refined that model with a low-score dependence correction and a time-decay weighting, and they framed the result as a test of betting-market efficiency. The Elo rating system, originally designed for chess (Elo, 1978), provides an alternative team-strength estimate that updates on match results alone and avoids the score-line modelling that Poisson approaches require.

Tournament-level forecasting has tended to publish one question at a time. Groll et al. (2019) combined a random forest with ranking and Poisson components to forecast match outcomes, then simulated the bracket to produce each team’s chance of the title in international tournaments. That design answers the champion question well, although it does not address a contest that pays for one combination of answers to different questions.

Bookmakers build a margin into their odds, so the probabilities the odds imply add up to more than one. Removing that margin to recover true probabilities is called devigging, and the simplest method, proportional normalisation, scales every implied probability down by the same factor. Shin (1993) derived a model that attributes part of the margin to informed traders, and Štrumbelj (2014) compared the methods on football and found the basic and Shin variants competitive. We use proportional devigging as the baseline and report how the result changes with the assumed margin.

The scoring rules used throughout are standard. The logarithmic score (Good, 1952) and the Brier score (Brier, 1950) are both proper, meaning a forecaster minimises the expected penalty by reporting true beliefs. We report log-loss as the primary metric because it punishes confident errors hardest, the failure that matters most for a single ticket.

4.1 Data sources

Team strength is estimated from the martj42 open dataset of international results (Jürisoo, 2024), which covers matches from 1872 to the present. Player scoring rates for the historical calibration come from the companion martj42 goalscorers dataset. The 2026 live ticket uses declared tier priors instead, because per-90 scoring data was unavailable, a substitution recorded in Section 7. The match calibration draws on 11,734 matches from 2014 onward and holds out the 1,767 played after September 6, 2024 as the test set (Tab. 1). Historical champion odds for the beat-the-line test are transcribed from the public outright-history tables of vegasinsider.com, whose free tables give odds only for the eventual champion and the documented pre-tournament favourite, a gap discussed in Section 7. The 2026 market comparison uses the devigged Polymarket outright winner market. All inputs are frozen to a content-addressed snapshot before kick-off, so no information from after kick-off enters the locked pick.

4.2 Team strength and the engine-selection inversion

Two team-strength models sit behind a common interface: an Elo system (Elo, 1978) and a Dixon-Coles Poisson model (Dixon & Coles, 1997). On held-out single matches Dixon-Coles is the better-calibrated model, with a log-loss of 0.867 against Elo’s 0.894, and it also has the lower expected calibration error (Tab. 1).

The contest scores the champion rather than individual matches, and on that question the two models swap places (Fig. 2). In a backtest of the 2010 to 2022 World Cups that uses only information available before each tournament, Elo scores a mean champion log-loss of 1.889, better than the market’s 2.199. Dixon-Coles scores 2.504, worse than both. Section 5.2 reports Elo’s champion log-loss as 1.897, because the market comparison is a separate backtest with its own random seed and comparison field; both round to 1.9 nats. Dixon-Coles over-rates South American (CONMEBOL) sides, because high-scoring qualifiers within that confederation inflate their attack ratings, and in a live run it rated Brazil far above the market, an artifact of that inflation. The bracket therefore uses Elo, the model that beats the market on the question the contest asks, and Dixon-Coles is kept for the per-match comparison it wins.

4.3 The bracket and the player models

The simulator reproduces the 2026 format exactly: twelve groups of four, the FIFA tiebreaker order, the eight best third-placed teams resolved through the encoded combination table, and the Round of 32 through the final with extra time and penalties. Each of the 50,000 simulations draws from a single seeded random number generator, so a given snapshot and seed reproduce an identical output. A player’s goals are drawn from a Poisson process whose rate combines his scoring rate with the matches he is expected to play. Matches played depend on how far his team advances in that simulation, which is what ties the scorer questions to the champion. The scoring rate is observed for the historical backtests and a declared tier prior for the 2026 ticket. It is recency-weighted with a one-year half-life, so a goal scored a year ago counts half as much as one scored today, in the spirit of the time-decay weighting of Dixon & Coles (1997). Section 5.5 reports the calibration gain behind that choice, and the designated-penalty-taker signal that was tested and rejected. The Player of the Tournament model scores candidates on team success, individual goal contribution and a star-prior term calibrated against the historical Golden Ball vote.

4.4 Scoring rules and the joint objective

The primary metric is the logarithmic score over realized outcomes, written for the realized-outcome probabilities as

where the hat-p term is the probability the forecaster assigned to the outcome that actually occurred. Lower is better, and a confident wrong forecast is penalised sharply (Good, 1952). Before the 2010 tournament the engine gave Spain 26.0 percent, which scored 1.348 once Spain won. Before 2018 it gave France 6.7 percent, which scored 2.697.

The pick is the argmax of the simulated joint frequency table,

where the probability is estimated by counting how often each complete ticket occurs across the simulations. Picking the four favourites separately optimises four copies of (Eq. 1) independently and ignores the links between the questions, so in general it does not return the argmax of (Eq. 2).

Market probabilities are recovered from decimal odds by proportional devigging,

where the denominator is the booksum, or overround: the sum of the implied probabilities, which exceeds one by the bookmaker’s margin. The historical comparison assumes an overround of 1.30 for the outright market, and the beat-the-line result keeps its sign for any value from 1.20 to 1.40. Shin (1993) and Štrumbelj (2014) describe margin models that reallocate the overround unevenly, which we leave as a refinement.

4.5 Software environment

The engine is a Python package, type-checked under mypy strict and covered by a full test suite. A hard calibration gate blocks any pick lock unless every blocking check passes on real evidence. The analysis recomputes every number this paper derives from the source datasets and regenerates each figure with pandas alone.

5.1 Match calibration

On the held-out test set, Dixon-Coles is the better single-match forecaster on all three metrics (Tab. 1). Both models clear the uninformed forecast, one chance in three for each result, by a wide margin: that forecast scores a log-loss of 1.099, against 0.867 for Dixon-Coles. An expected calibration error of 0.037 indicates a forecast whose stated probabilities track realized frequencies closely.

Table 1. Held-out match-outcome calibration on 1,767 international matches played after September 6, 2024. Lower log-loss and ECE are better.

5.2 The model beats the closing line on the champion

The engine beat the betting market on the champion in three of the four World Cups from 2010 to 2022 (Fig. 1). In 2022 it gave Argentina a 24 percent chance before the tournament, nearly three times the 8 percent in the devigged closing odds. The one loss was France in 2018, at 7 percent against the market’s 12. Over the four tournaments the engine’s mean log-loss was 1.897, against the market’s 2.199. Those are the scores of a forecaster who gives every eventual champion about 15 percent and of one who gives it about 11. The champion is the one contest question with a liquid market to test against, which makes this comparison the closest thing available to an out-of-sample audit of the engine.

5.3 The goal-count band

The exact goal count carries real spread, and the model locates it well. Every Golden Boot from 2010 to 2022 was won with five to eight goals, and in each of those tournaments the model put more than half its probability on that range (Fig. 3). Its share rose from 51 percent in 2010 to 84 percent in 2022 and averaged 62 percent. Which player tops the scoring chart is the larger source of joint uncertainty, the subject of Section 5.5.

The Player of the Tournament model is calibrated against the historical Golden Ball vote with coefficients alpha 0.5, beta 2 and gamma 4, the last reflecting a media electorate biased toward star players. In all four tournaments the Golden Ball winner ranked in the model’s top five, and Messi was its first choice in both years he won (Tab. 2).

Table 2. Player-of-the-Tournament calibration against the historical Golden Ball winner, by tournament. The realized winner ranked inside the model top five every year and first in both Messi years.

5.4 The 2026 pick and where it disagrees with the crowd

For 2026 the engine rates Spain and Argentina higher than Polymarket’s devigged prices do, and France, England and Portugal lower (Fig. 4). The two largest gaps are about 8 points each. Spain is the engine’s favourite at 24.0 percent, against 15.9 percent on Polymarket, and Argentina’s gap is as wide, at 16.8 percent against 8.5.

The selected ticket is Spain as champion, Erling Haaland as both Player of the Tournament and top scorer, and seven as the exact goal count (Tab. 3). It wins with a probability of 0.94 percent, a little under one chance in a hundred, and the simulation’s 95 percent confidence interval runs from 0.86 to 1.03 percent. A casually built entry has a chance of 0.05 to 0.10 percent, one in a thousand at best, so the pick is roughly ten to twenty times as likely to win. The goal count is a near-tie: the single likeliest ticket has eight goals, at 0.95 percent. The pick takes seven instead, the count fewer entrants are expected to choose, because a shared pool pays each winner more when fewer entrants split it. The other alternatives in Table 3 each change one part of the pick.

Table 3. The selected 2026 ticket and its nearest alternatives, from a frozen run of 50,000 simulations (snapshot c7f3f5afa805, seed 20260611). Joint P is the simulated probability that all four components resolve correctly. The pick tilts off the strict argmax (eight goals) to seven, the less public count.

5.5 Weighting recent form opens the Golden Boot race

Who finishes top scorer is the largest single uncertainty in the ticket, and weighting recent goals more heavily made that race more open. When every career goal counts equally, the favourite for the 2026 Golden Boot holds 28.5 percent of the probability, and with recency weighting he holds 23.8 percent (Fig. 5, Tab. 4). Counted as the number of equally likely contenders that would spread the probability as evenly, an inverse-Herfindahl measure, the field widens from 7.8 to 11.4. The five leading names together fall from 66.7 percent of the probability to 55.6 percent. No ticket can be likelier than the chance of the scorer it names, so the flatter race lowered the best ticket’s probability from 1.02 to 0.95 percent. Better modelling of current form lowered the headline number because it showed a Golden Boot race more open than career goal totals suggest.

Table 4. Concentration of the 2026 top-scorer marginal, career-volume baseline against the integrated one-year recency model. A higher effective field and a lower modal share mean a more open race.

The weighting was adopted because it forecasts past Golden Boots better. Over the realized Golden Boots of 2010 to 2022, a one-year half-life cut the mean scorer log-loss from 12.656 to 12.051, the best of the half-lives tested (Tab. 5). It also raised the chance the model gave the eventual winner in the two most recent tournaments (Fig. 6). Kane’s chance in 2018 nearly tripled, from 0.7 to 1.8 percent. Mbappé’s in 2022 nearly doubled, from 1.7 to 3.2 percent, while neither version gave Müller, the 2010 winner, any chance. A designated-penalty-taker signal moved calibration the wrong way and was rejected, because it entrenches a team’s established penalty taker over the breakout scorer who tends to win the Golden Boot.

Table 5. Recency-weight tuning for the scorer model, mean scorer log-loss over the 2010 to 2022 realized Golden Boots. Lower is better; the one-year half-life was adopted.

The pick comes from counting complete tickets across the simulations, and that count shows where the uncertainty sits. Four of the five likeliest tickets share Spain, Haaland and Haaland and differ only in the goal count (Tab. 3), so once the champion and the scorer are fixed, what remains open is the count. An entry built from four separate favourites could land on a combination the simulations rarely produce together, and reading the pick off the table of complete tickets avoids that failure by construction. The top scorer is the binding term: even the favourite holds only 23.8 percent after recency weighting (Fig. 5), and that ceiling is why the headline sits below one percent.

The beat-the-line result gives the champion an external check that the other three questions lack. A model that beats a liquid, devigged market on the most-traded question over four tournaments has shown more than internal consistency (Fig. 1). The same engine then carries the player questions, where no comparable free market exists, so their calibration evidence rests on the goal-count band (Fig. 3) and the Golden Ball rank history instead of a market test.

The 0.94 percent is a real edge of small absolute size, on a sample of one. Ten to twenty times a casual entry is a large edge in relative terms, and the ticket still loses about 99 times in 100. On a single ticket in a single tournament, chance dominates, and no result here should be read as a prediction that the ticket wins. The better-calibrated scorer model lowered the headline from 1.02 percent rather than raising it, which is the calibration working as intended: a more accurate scorer forecast made the ticket’s estimated chance smaller.

1. The 48-team bracket has no precedent. No World Cup has been played in the 2026 format, so the component models can be backtested but the bracket structure cannot. The historical tests in Section 5 use the 32-team format, and the path probabilities for the new structure are calibrated only indirectly, through the market anchor and internal consistency.

2. The Golden Boot market is unverified. No free historical market for the top-scorer count was found, so that part of the edge rests on the goal-count calibration rather than on a test against a price. The contest’s player questions therefore carry weaker external validation than the champion question does.

3. Player rates depend on example priors. Per-player scoring rates come from declared tier priors, because live per-90 data was unavailable. A real entry would replace those priors with observed rates and check every candidate against the final squads. Recency weighting (Section 5.5) improves the scorer model’s calibration, and the identity of the top scorer remains the largest residual uncertainty in the ticket.

4. The historical sample is four tournaments. A mean log-loss over four World Cups cannot separate a stable edge from a fortunate draw with high confidence. The result is suggestive and points the right way, and it is not a large-sample claim.

5. The market anchor is champion-only and thinly sourced. Historical outright odds were available for the champion and one favourite per year, not the full field, and the comparison assumes a 1.30 overround. The sign of the result holds across plausible overrounds, while the exact margin does not.

A contest that pays only for all four answers together rewards the likeliest combination, and the engine picks that combination instead of four separate favourites. One external check earns it trust: its champion forecasts beat the devigged closing line in three of the last four World Cups. Their mean log-loss was 1.897, against the market’s 2.199. The goal count is well located, with 62 percent of the model’s probability on the five to eight goals that every Golden Boot from 2010 to 2022 fell within. The identity of the top scorer is the larger unknown: even the favourite holds only 23.8 percent once the scorer model weights recent form.

The selected 2026 ticket is Spain, Haaland, Haaland and seven goals, at a joint probability of 0.94 percent. That figure is a defensible model estimate of a small chance on a sample of one, roughly ten to twenty times a casual entry’s. Integrating a better-calibrated scorer model lowered it from 1.02 percent, a more accurate estimate rather than a weaker ticket.

The next iteration should replace the example player priors with observed scoring rates and source a wider historical field for the market comparison. It should also pre-register the 2026 ticket before kick-off, so the single live outcome becomes a small but clean addition to the calibration record.