# Tiebreaker options in the Slam tournaments

Hi all,

The Slam tournaments, which are part of the Memory League World Tour, will involve multi-set matches (similar to a tennis match). We need some way to choose a discipline as a tiebreaker when the score in a set is 4-4.

The ideas we’ve had are explained in this document (huge thanks to @SimonReinhard and @Sylle for creating it):

tiebreaker_options.pdf (445.4 KB)

We’d be happy to hear your thoughts!

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Basically, three different concepts:

Option I: Random choice between disciplines chosen by both competitors during the set
Option II: Choice between disciplines chosen by both competitors during the set using vetos
Option III: Choice based on ranked lists of preferences, regardless of choices during the set

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Spontaneous thought: with Option III it seems possible to become a “mindgame”.

Let’s assume a scenario where the strengths/weaknesses/preferences of the two athletes are the complete opposite (which is not that unlikely actually), e.g:

If player A simply switches preference order for his first two choices, then the chosen discipline would be Numbers. Of course, Player B can counter this by either (1) switching preference order for their first two choices or (2) switching preference order for their last two choices.

If they both do (1), we will get a random event among two disciplines that are even more uneven than before (in this case, Numbers or International Names):

I will not elaborate too much on the other possible scenarios here (different combinations of player A and B doing (1) and (2) respectively), but it seems to me that this system can be “gamed”? Or am I missing something?

Option I and II look nice. Option II seems to be the most fair imo.

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Full disclosure: I’m the one pushing for Option III. I’m not sure if I’d use “gamed”, maybe rather “leads to interesting strategies”!

One detail which wasn’t mentioned is the possibility to have a filter on the delta in order to remove the disciplines that have a delta > 3 (which is actually only a small subset of the cases for the highest sums). In your second scenario, it’d become a random choice between Names and Images.

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Hello everyone !

And thank you to the many masterminds working on these options and rules.

I don’t like the option II because of how it “feels” to be vetoed or to veto someone else’s choice.

I really like option I because of the randomness and surprise it creates. (It also pushes people to train every discipline and to be ready for anything)

I like option III as well because of the strategic aspect of it and how people can start worrying and planning 3 weeks ahead…

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@guillaumepetitjean Thank you!

Maybe the term “veto” (which also appears in the PDF, admittedly) is a bit misleading for Option II. How it would work, according to my understanding, is that the players would be presented with the three (or four) overlapping disciplines = the possibilites for the tiebreaker events. Then player A could “scratch off” one discipline, Player B would do the same. Then either one discipline remains and is played (when starting with 3) or two remain and random choice decides (when starting with 4).

So, the whole process of “a player chooses an event and the other player vetoes it”, which was part of the Remote Tournament rules, does not even happen here: nobody chooses anything in the first place but the players only scratch off a discipline each.

Personally, I see something to like in all three options. When pressed, I would probably also trend towards Option I because of its simplicity, but all three seem fine :). .

Hello everyone, nice ideas!

I would say, Option II is the fairest one. Option I is interesting because it could add an element of surprise/randomness and could increase the chances of the underdog.
At first I thought Option III is too complicated, but there is one thing that I dislike about Option I and II (and of course I am biased here): I believe there would never be a decider in Names or International Names. If for example Player A chooses National Names, then in most of the cases Player B would choose International rather than National Names. And then we treat them as different events, so there will be no overlap. So it is an advantage of Option III that every discipline has the chance to become the deciding event.

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@JHB I think when using Option III, the discipline a player ranks as their least favorite one (rank 6) is mathematically excluded from being chosen.

And the next-to-least favorite one (rank 5) would be chosen very very rarely because the cases that present the highest opportunity for it (= if the other player ranks that exact event as his most favorite one, rank 1) would be excluded via the formula because they would have a delta of larger than 3 (5 - 1), if I got @Sylle correctly.

So, I think at Option III some events do not have any chance of being chosen (those either player has on rank 6) and for some disciplines it would be extremely unlikely to be chosen (those on rank 5) due to the “delta not larger than 3” rule. In a way, at Option III each player has a full veto and a “quasi-veto” if they want to exclude events with 100% (rank 6) or with very high probability (rank 5).

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@SimonReinhard Yes, true. But there are excluded from being chosen because of someones wants to veto it. My point is that in Option I and II there will never be an overlap of a Names discipline, because the weaker player in Names can always choose the other Names discipline.

Lets say we a have a situation where the strengths/weaknesses/preferences of the two athletes are the complete opposite (from Florians example)
Player A vs. B

Player A chooses Int. Names, Words, Images and wins all of them
Player B chooses Cards, Numbers, Images and wins all of them
In Round 4 Player A has to choose Cards and loses
Player B can now choose Words or Names. Although he might have the higher chance of winning in Words, it could be better to choose Names because then the tiebreaker is chosen between Cards and Images and not between Cards, Images and Words. So the players are the complete opposite, but Player B is favoured by the tiebreaker-rule.

That is what I don’t like about Option I and II and therefore would prefer Option III.

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I adore this, combining my loves of game theory and Memory League Here are my thoughts.

The Connection Between Player’s Preferences

One player’s gain in this decision is always another players loss. Given that each player can look up the other’s scores, both of them will develop a close to perfect understanding of which disciplines they are most likely to win. Indeed, both players could come close to ranking how likely they are to win each discipline.

Of course, this will not be perfect. Players will imperfectly estimate both their own and their opponent’s abilities. Besides which, a player might want discipline which they know isn’t the perfect choice, simply because said discipline makes them feel more comfortable. Nonetheless, if both players ranked all 6 disciplines truthfully in terms of their preferences against their given opponent (producing a list called their ‘preference relation’) I believe the two rankings would be almost exactly the reverse of each other

Effect on the Given Options

This connection between preference relations clearly underlines the advantages of Option I and Option II. Let us assume that two players, A and B, have reached a tie break. The preference relation for A is as follows:

1. Numbers (100% chance win)
2. Cards (80% chance win)
3. Images (60% chance win)
4. Words (40% chance win)
5. Names (20% chance win)
6. International Names (0% chance win)

We will assume that B’s preference relation is the reverse of this. This means A will choose Numbers, Cards, Images and Words, while B will choose International Names, Names, Words and Images, thus for both Option I and Option II. (For simplicity’s sake I will assume that someone can choose both Names and International Names here, but I will come back to this.) Thereby, for both Option I and Option II, the discipline chosen in the tie break should supposedly be be a coin flip between Images and Words.

Worked Example

@JHB has written the key point of which which indicates why neither Option I nor Option II are viable. In a perfect world, where competitors could choose both Names and International Names, this flaw shouldn’t be revealed, but one player cannot choose both Names and International Names in the majority of competitions.

(This is an adaptation of JHB’s example, credit should go there. However even if you’ve already read that, this is still worth reading for both the maths and the explanation of why that still applies in Option II scenarios).

Playing Under Option I

Imagine we are playing a game under the rules of Option I, with the winning probabilities for A as outlined above. Imagine it is A’s turn to choose. A has already won their own choices of Numbers, Cards and Images, while B has won their choices of International Names, Words, and Images but had already lost their choice of Cards.

If A chose to play Names, A would win that individual game only 20% of the time, but win the tie break (which would be mixed randomly between Images and Cards) 75% of the time. Scaling this for the probability that the game goes to the tie break. This means A would win before the tie break 20% of the time, and in the 80% of games when it went to a tie break, they’d win 75%, lose 25%. Thereby, A would win 20% + (80% x 75%) = 20% + 60% = 80% of the time when they chose Names.

If A chose to play Words, the option they’re better at, they’d win 40% of the games before the tie break. If the game got to the tie break they’d face an equal chance of Cards, Images and Words, which they’d win 80%, 60% and 40% of the time respectively, thus they would win the tie break 60% of the time. Thereby, A would win 40% + (60% x 60%) = 40% + 36% = 76% of games when they chose Words.

Playing Under Option II

Think through the exact same scenario under Option II rules, when each player has a right to veto.

If A chose to play Names, it would play out exactly as above. They’d win only 20% of games before the tie break, but in the tie break, they would play half of the games in Cards, half in Images, which A would win 75% of the time, meaning if they played 100 games, you’d expect them to win 20 before the tie break, 60 after the tie break, and lose 20 in the tie break, meaning they’d win 80% of games.

If A chose Words, both sides would have a chance to veto in the tie break. Thereby, of Cards, Images and Words, A would veto Words, B would veto Cards. As A would still have a 60% chance of winning in the tie break, the odds are as before, A would win 76% of games rather than 80%, meaning A would lose out by choosing their preferred category.

Option III

Sadly, this doesn’t mean that Option III is the best option. Assuming, as before, that one player’s preference relation is the reverse of another player’s preference relation, there is an obvious problem with Option III.

Imagine this is A’s preference relation:

1. Numbers (100% chance win)
2. Cards (80% chance win)
3. Images (60% chance win)
4. Words (40% chance win)
5. Names (20% chance win)
6. International Names (0% chance win)

Thus, this is B’s preference relation:

1. International Names (100% chance win)
2. Names (80% chance win)
3. Words (60% chance win)
4. Images (40% chance win)
5. Cards (20% chance win)
6. Numbers (0% chance win)

If A and B wrote this honestly, then using the Borda count (5 points for first choice, 4 points for second etc) as Florian points out, every option will get 5 points. Awarding delta values (effectively a penalty for placing a choice higher up) would allow us to separate the 6 options down to 2 (Words and Images) but that assumes nobody is gaming the system.

The system isn’t difficult to game either. If B was honest, and A simply swapped Cards and Numbers in their preference relation, the total points for each category would be Numbers - 4, Cards - 6, Images - 5, Words - 5, Names - 5, International Names - 5.

Obviously, B can game the system just as much as A can. Indeed, you could argue that both sides have an equal advantage when it comes to lying about their preferences, thereby Option III is the fairest option of them all. I respectfully disagree. If you enter that effective lottery, and one player ends up with their second choice, which the other player naturally views as their fifth choice, then we are no longer playing a game of memory. The winner will believe they’ve bested the other at a subgame of skill, the loser will believe they’ve been unlucky. Likewise, if it was a purely random choice between all 6 options, one person will be hard done by by a random number generator

To me, it comes down to your perception of the word ‘fairness’. Personally, choosing one of the middle two categories (in our example Words and Images) is the ultimate fairness. One person may have an advantage in both of these categories. To me that’s fine, it means they’re a better player. If both players have a big advantage over the other between the middle two categories, (i.e. A would win 90% of Images games, B would win 90% of Words games) then there’s nothing you can do about that. One player is good at half the games, the other is good at the other half, they’re never going to agree, one will always have a big advantage.

I would, thereby, like to put forward a 4th option

Option IV

This option is unexciting. But I believe it’s easy to understand, impossible to game, and will result in one of the two ‘fair’ outcomes being chosen.

All disciplines are put on the table.Players take it in turns to pick 2 disciplines that they don’t want to play. Then, they see if they agree on which of the 2 remaining disciplines they would rather play. If they choose the same one, they play it. If not, it goes to a coin toss.

If the players don’t try to game the system, here is what will happen in our worked example:

A vetos International Names. B vetos Cards. A vetos Names. B vetos Numbers. A offers to play Images. B chooses a coin toss between Words and Images instead.

If A tried to game the system, for instance by choosing to veto Words rather than Names, B will still choose the coin toss rather than Images. This time though, the options are worse for A. They might still play Images, but in trying to cheat the system, all they’ve done is open up the possibility that they’ll play words.

I understand if you’d rather go for one of the ‘gamier’ options, but Option IV is easy to understand, quick, always fair and impossible to game

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Hi! Nice analysis. Need to get into the finer details but sounds very interesting.

It has been mentioned by some that vetoing another one’s choice might be seen as too “destructive” or unpleasant. Personally, I think it is not such a big problem, but to accomodate those people, too, maybe one could mirror your Option IV:

Step 1: Calling events

Player A chooses two events they want to play.
Player B does the same, without knowing player A’s choices.

Now there are two possibilities: Either the players have chosen at least one identical event - for that see step 2a. Or they have chosen four different events - see step 2b.

Step 2a: Identical event(s)

If one event is identical, it is played.

If two events are identical, each player says another event from these two. If they say the same, it is played. If not: Random choice.

Step 2b: No identical event(s)

If the players have said four different events, then two events are left. From these two, each player says another event they want to play. If it is the same, it is played. If not: Random choice.

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Yes, very nice analysis @GildedTruffle.

I like Option IV. And I would prefer Option IV in the “Veto-variant” , I also don’t see the problem with the “destructive” veto.
@SimonReinhard Regarding the other way of Option IV: It would be a mindgame again: If you are sure, that your opponent will pick his two best disciplines, then it would be best to choose the two middle ones, because then there would be a random choice between your two best ones. Of course, one could argue, that it is just smart and strategic play then, but I would rather try to avoid such situations. In the Veto-way we can be sure that everyone really vetoes the discipline that he does not want to play.

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I looked at my spreadsheet again, with a limit on the delta set at 3 (deltas of 4 not used), the choices #5 and #6 are never picked in the end.

There’s no honesty / dishonesty there, the goal for each player is to maximize the amount of points for the disciplines they’re most willing to play, not in a vacuum, i.e. when actually considering what the other is most likely to place high on their list.

Hi Sylle,

Sort of. Technically the ‘honest’ strategy should be referred to as ‘the strategy in which the player ranks the disciplines in the same order they appear in their preference relation’. Indeed I was only mentioning the ‘honest’ strategy to refute it. However, with your suggested delta limit of 3 it does become it does become slightly more viable again, if easy to manipulate.

Delta Limit 3 Option III

Just to establish, the delta is the difference between where the two players ranked a discipline. For example, if A said Names was their 1st choice and B said Names was their 4th choice, the delta would be 3. With the recommended delta limit of 3, it would not be eliminated, since the delta of 3 is not greater than 3.

Yes. For both players, neither their 5th nor 6th choice disciplines will ever be chosen. This means that options 5 and 6 effectively act as vetos. Each player has two vetos. From there, they’re only deciding between the final two non-vetoed disciplines. Sound familiar? However, it isn’t quite truth revealing like Option IV.

For example, let’s imagine, as before, that A is playing against B. A’s preference relation is, as explained before, the reverse of B’s, and as follows:

1. Numbers (100% chance win)
2. Cards (80% chance win)
3. Images (60% chance win)
4. Words (40% chance win)
5. Names (20% chance win)
6. International Names (0% chance win)

Imagine A chooses this as their option ranking. If B names Words as their first choice option, Words will get 7 points. If B places their remaining options in reverse preference order (i.e. plays the strategy Words, International Names, Names, Images, Cards, Numbers) then Words will be selected as the tie break option, not Images.

Personally, I’d rather have a system which you can’t game at all. I explained why in my previous post. On top of that, every match done in this system will both need an adjudicator and need to be explained very precisely to any new competitors/audience members. However, if you want the theatre of a gameable system while allowing each player could guarantee that they don’t get one of their two least favourite disciplines, then Option III with a delta limit of 3 is an excellent option.

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Partly true, since one strategy could (will?) be to veto one’s own best discipline when you “know” the opponent will veto it anyway. This way you have more points to put on other disciplines which you opponent and yourself are maybe more keen on playing.

I respectfully disagree, the only thing you need to inform the competitors is that they have 15 points to assign to all 6 disciplines in the form of 5+4+3+2+1+0 and that the discipline with the highest sum is chosen. That and the last resort is to randomly choose between the two remaining disciplines. The limit on the delta can be named as a way of preventing some kind of level of objective unfairness. My point is that with other systems, the competitors (sometimes not speaking/understanding English very well) need to engage in some kind of discussion with the arbiter or opponent. I feel that it’s easier for there to be misunderstandings there than with option III. With option III, you know the rules beforehand, you can think of a strategy (because the strategical aspect is key) and when the time comes you just write your list, submit it and the computer does the rest.

I think I like option IV better than option I and II, but am afraid again that there are many steps and interactions and risks for misunderstandings (this is in part based on the occasionally very creative interpretations of the current rules by League competitors).

Option V:

Taking the discipline which, during the previous set, ended up with the closest result in term of points (as calculated based on the IAM levels).

(probably too biased towards names and words though …)

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I think a number of factors need to be taken into account when it comes to choosing the “best” / most fitting option:

A. Simplicity

It should be easy to understand. There will be language barriers and there will be athletes who do not speak English.

For Option I there would be no interaction necessary between the arbiter and the players apart from the arbiter communicating the name of the tiebreaker event, so it seems the simplest.

Option II seems easy to understand and would require either no interaction by the players (if two events overlap) or would require a single veto by each player. The arbiter would communicate the tiebreaker event.

Option III, while clear enough once one has read its rules, seems to have the highest complexity and thus the highest potential for misunderstandings. This would apply to doing Option III “per hand” and for an automatic Option III. Also, the finer points of strategy might elude players with a language barrier.

Option IV seems easy to understand and straightforward once the concept of vetoeing has been understood.

I would personally rank the Options in terms of simplicity as follows:

1. Option I
2. Option II and IV
3. Option III (by hand and electronic version)

B. Speed of Execution

If a player wants to play their tiebreak game quickly, this should be respected. I think it would not be ideal if choosing the tiebreaker takes a few minutes, potentially breaking one player’s rhythm.

In that regard Option I surely is the fastest: The arbiter will know the overlapping disciplines before match 8 is even played and can present the athletes with the randomly chosen tiebreaker event right after match 8 is over.

Option II is faster than Option IV for two overlapping events. For three and four overlapping events, Option II and IV seem to take about the same time.

Option III seems the slowest if done by hand and transmitted to an arbiter and might be roughly as fast as Option II and IV if done electronically, but even then slower than Option I.

I would personally rank the Options in terms of speed as follows:

1. Option I
2. Option II and IV (Option III - electronic version)
3. Option III (by hand)

C. Accuracy of Execution

An Option needs to have as few pitfalls and as little room for errors as possible.

Option I: Since the overlapping events are very easy to determine, errors most probably will not happen with a properly trained arbiter.

Option II and IV: Both seem straightforward enough so that errors by the arbiter in conducting the options should not happen.

Option III: If Option III is done by hand, I see quite some potential for errors with summing up things and comparing deltas. Errors might be noticed ultimately but correcting them would take additional time. With an electronic Option III, the potential for errors would be very small, of course, and nil if there are no bugs.

I would personally rank the Options in terms of accuracy as follows:

1. Option I (Option III - electronic version, bug-free)
2. Option II and IV
3. Option III (by hand)

D. Fairness

A fair option is an option where no player has an inherent advantage over a large number of games. In that regard, Option I - IV seem fair to me.

Option I and II have the basic underlying rationale that you cannot really complain about any discipline that you have chosen yourself in the match. Any kind of potential information advantage that the player who chooses second might (supposedly, as stated in this thread) have will be canceled out over a larger number of matches because the order of choice will change.

Option III seems fair in that regard that each player has the same chance of “gaming the system” and thinking about complex “if they do this, I do this, and if I do this etc…” strategies. But players new to the game or players not familiar with the strategies might be at a bit of a disadvantage here.

Option IV certainly seems fair because it is very transparent and does not seem to grant any advantage to the one who has the first choice of veto.

Personally, I think despite some minor differences each Option is equally fair.

I will not sum it up now according to how the events were ranked in each category because the categories might have different weight according to different people, so this might simply serve as a basis for discussion.

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But but but, strategy and fun?

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Sorry, reading through my replies I was probably at least on the verge of being trollish … my apologies.

I would make a difference between “complexity” and “strategical aspect”. Saying “the highest sum takes it” is pretty much the whole story from a technical point of view. You don’t need to explain to everyone in every language the subtilities of strategy, since the possibilities are too many to mention and that you can’t anticipate everyone’s strategy for them. The technicalities of the voting itself are however very simple.

One advantage, as I see it, is also that it makes the tiebreaker a different phase, separate from the set itself. During the set, each game is pretty much chosen with the goal of maximizing one’s chances of winning said game. There I feel it becomes too complex to need to think about one’s choices and the effect they might have on the tiebreaker choice at the same time. It feels cleaner to start a sort of negociation process only once one has reached 4-4.

So yes, I would add a category E. to the four you mentioned: Strategy, and not have “Strategy” being a bad word (akin to “gaming the system”), rather an integral part of the game.

I would love to hear more from more athletes who will potentially have to deal with this, i.e. players who are likely to participate in the Open tournaments (maybe something like the top 30 current players, whatever this may mean?).