Abstract

The 2024 revision of the FIDE tie-break regulations introduced an iterative formulation of the Direct Encounter criterion for Swiss-system tournaments, allowing its application to partially connected tied groups. While this refinement resolves known limitations of the classical formulation, we show that its iterative application exhibits a structural instability.

Using a minimal formal model based on weighted directed graphs, we prove that repeated reapplication of Direct Encounter after removal of the subgroup winner may violate restriction consistency: the relative order of two players can reverse solely as a consequence of deleting results involving a third player. The phenomenon is derived algebraically and illustrated with a concrete numerical example.

The mechanism identified here reveals a general instability inherent in iterative restriction-based ranking procedures and is not specific to the chess context.

Introduction

The 2024 revision of the FIDE tie-break regulations introduced a refined formulation of the Direct Encounter (DE) criterion (C.07, Article 6)[^1], allowing its application even when not all players in a tied group have played each other. This development resolves well-known limitations of the previous formulation in Swiss tournaments.

The Classical Three-Player Case

Before turning to the formal analysis, it is useful to recall the classical three-player situation that motivated the 2024 refinement of Direct Encounter. Consider three tied players A, B and C in a Swiss tournament. Suppose that:

• Player A defeated both B and C,
• Players B and C did not play each other.

Under earlier formulations of Direct Encounter, the criterion could not be applied because the mini-round among tied players was incomplete. However, from a logical perspective, A is clearly the strongest player within the tied group, since both B and C lost to A.

The revised formulation of Article 6 correctly addresses this case by allowing the clear subgroup winner to be determined even when some internal games are missing.

The structural issue examined in this note does not concern this improvement. It arises only when Article 6 is applied iteratively — removing the top-ranked player of a tied group and reapplying the procedure to the remaining players — a structural effect emerges that deserves examination. The objective of this note is to analyze that effect in a minimal formal framework. Mathematical formalism is kept to the minimum necessary for clarity, with references for readers wishing to explore the broader literature.

Minimal Formal Model

A Swiss tournament can be represented as a directed weighted graph:

• vertices represent players;
• directed edges represent games;
• weights $w_{ij} \in {0, \frac{1}{2}, 1}$ denote the score obtained by player $i$ against player $j$.

Let $S$ be a set of players tied on total score.

The Direct Encounter score of player $i \in S$ is:

$$ DE_S(i) = \sum_{\substack{j \in S \ j \ne i}} w_{ij}. $$

Players are ranked by decreasing $DE_S(i)$. The iterative formulation proceeds by:

1. ranking players in $S$ according to $DE_S$;
2. removing the highest-ranked player (and all results involving that player) from the set;
3. reapplying the same procedure to the reduced set;
4. repeating until all players are ranked.

Restriction Consistency

In ranking theory and social choice theory, stability properties under modification of the set of alternatives have been widely studied (see, e.g., Sen 1970; Young 1974; Saari 1995; Laslier 1997).

The idea considered here is related to what may be called consistency under restriction, namely the stability of a relative order when the set of alternatives is reduced.

Although terminology varies across the literature, the underlying principle is that relative comparisons between two alternatives should not depend solely on the presence of a third, removed alternative.

In practical terms, if player $i$ ranks above player $j$ in a set $S$, removal of a third player should not reverse their relative order. More precisely, let a ranking method produce an order $\succ_S$ on a set $S$. A method is restriction-consistent if, whenever $i \succ_S j$, then for every reduced set $S' \subseteq S$ containing both $i$ and $j$, one also has $i \succ_{S'} j$.

This property is not universally satisfied by ranking systems. Nevertheless, it is a natural structural expectation: the relative order between two remaining players should not reverse merely because a third player is removed, unless new information is introduced.

In the iterative application of Direct Encounter, no new information is introduced when a player is removed; existing information is deleted. Therefore, any inversion in the relative order of remaining players must arise from the structural effect of that deletion.

A Concrete Counterexample

We now illustrate the structural effect with a concrete example extracted from a Swiss tournament.
Table 1 shows a tied group of players with equal total score.

Step 1 — Determination of first place

From Table 1, Player #2 has Pts = 4.5 and max = 4.5. Therefore no other player can reach or exceed his internal Direct Encounter score. The first-place determination is straightforward and coherent: Player #2 is the clear winner of the Direct Encounter subgroup.

Step 2 — Removal of the winner

The iterative procedure (Article 6.3) now removes Player #2 together with all his results before reapplying Direct Encounter to the remaining players. Importantly, at this stage no new information is introduced: the method only deletes existing results.

Step 3 — The “hinge” that triggers the inversion

Consider Players #1 and #3:

• Player #1 drew with Player #2, hence $w_{1,2} = 0.5$ against the subgroup winner.
• Player #3 lost to Player #2, hence $w_{3,2} = 0.0$ against the subgroup winner.
• Player #1 defeated Player #3 in their direct encounter, i.e. $w_{1,3} = 1.0$.

After removing Player #2, Player #1 loses the half-point scored against the strongest player of the group, while Player #3 loses nothing comparable. As a consequence, when Direct Encounter is recalculated on the reduced set, Player #3 may rank ahead of Player #1 despite having lost the direct encounter.

Interpretation

This inversion is counterintuitive for players and difficult to justify in practical disputes because it is not caused by additional results or new games. It arises solely from deleting the subgroup winner and, with him, deleting a result that was favourable to Player #1.

In the terminology of the previous section, this constitutes a violation of restriction consistency: the relative order between two players may reverse solely because a third player is removed.

Structural Explanation of the Inversion Effect

The inversion observed above is not accidental; it follows directly from the algebraic structure of the iterative procedure.

Let $S$ be the original tied group and let $L$ denote the subgroup winner determined in the first step of Direct Encounter. For any remaining player $i \in S \setminus {L}$,

$$ DE_S(i) = \sum_{\substack{j \in S \ j \ne i}} w_{ij}. $$

After removal of player $L$, the new Direct Encounter score is

$$ DE_{S'}(i) = \sum_{\substack{j \in S' \ j \ne i}} w_{ij}. $$

Since the only removed contributions are those involving $L$, we have the relation

$$ DE_{S'}(i) = DE_S(i) - w_{iL}. $$

Thus, for any two remaining players $i$ and $j$,

$$ DE_{S'}(i) - DE_{S'}(j) = \big(DE_S(i) - DE_S(j)\big) - \big(w_{iL} - w_{jL}\big). $$

Even if $DE_S(i) > DE_S(j)$, the relative order may reverse after removal of $L$ whenever $w_{iL} > w_{jL}$. In words: if player $i$ scored better than player $j$ against the subgroup winner, then deleting that winner penalizes player $i$ more than player $j$, potentially reversing their order.

This structural feature may be described as a best-result elimination effect: the iterative mechanism may eliminate, for some players, one of their strongest comparative results (e.g. a draw or win against the strongest player of the group).

Practical Implications and Concluding Remarks

The example above does not suggest that Direct Encounter is fundamentally flawed. The first-stage determination of the subgroup winner is coherent and intuitive. The structural issue concerns exclusively the iterative reapplication after removing the already-ranked winner.

From a practical arbitral perspective, such inversions may be delicate in tournaments where titles, norms, or prizes are at stake, because they are not caused by additional results but solely by deletion of favourable outcomes against the subgroup winner.

A possible regulatory consideration would be to apply Direct Encounter only once for determining the top-ranked player within a tied group, and then proceed to the next listed tiebreak criterion for the remaining players. This would preserve the improvement introduced in the 2024 revision while avoiding the structural instability associated with repeated restriction.

While motivated by a specific regulatory context, the structural phenomenon identified here has broader implications for the theory of iterative ranking procedures.

References

• FIDE (2024). Tie-Break Regulations. FIDE Handbook, C.07 (Article 6.3). Available at: https://handbook.fide.com/chapter/TieBreakRegulations082024
• Sen, A. (1970). Collective Choice and Social Welfare. Holden-Day.
• Young, H. P. (1974). “An axiomatization of Borda's rule.” Journal of Economic Theory.
• Saari, D. G. (1995). Basic Geometry of Voting. Springer.
• Laslier, J.-F. (1997). Tournament Solutions and Majority Voting. Springer.

[^1]: FIDE Handbook, C.07, Article 6: https://handbook.fide.com/chapter/TieBreakRegulations082024