# Is it correct to infer a second, third etc place in board games with a score?

This may stray a bit into philosophy, but I believe it has some real-world impacts.

Many games rank players during the game, and some leave players in some sort of order at the end (ie. a 'score'). An example of this is Settlers of Catan, or Pandemic: Contagion Each player has a number of points an any time, and at the end of the game, the person with the most points is declared the winner.

A lot of rulebooks have wording like this:

The person with the highest score overall wins.

Pandemic: Contagion

If you have 10 or more victory points during your turn the game ends and you are the winner

Settlers of Catan

At the end of the 10th game round, the House that controls the most areas containing either a Castle or a Stronghold is declared the winner. If, at any time during the game, a player controls seven such areas, that player immediately wins the game.

A Game of Thrones. The Board Game

These phrases seem to indicate that there is only a winner, and the remainder all lose.

In other games (semi coopoerative games) there is sometimes no winner Semi cooperative games In such games, it is strongly implied that when the group loses, there is no 'second place' etc.

Lots of people I have played with sometimes infer that there is an implicit second place due to the fact that at the end of the game players are still ranked by score.

So there are two aspect to this that I think are slightly related: - Is it correct to infer positions beyond winner based on these scores? - Does doing so affect gameplay?

Is it correct to infer positions beyond winner based on these scores?

There are probably two ways of looking at this

1. At the end of the game, you take all the markers off the scoreboard, and put them in two piles: The Winner, and everyone else.
2. The other way is to say that the person with the second most points did 'second best' at the game.

I would argue that the 1st interpretation is correct for the victory condition given in Settlers of Catan, given its wording. Other games are maybe less clear. Some rules books explicitly say whether there are positions beyond 'winner'.

Interpretation 2. makes some questionable assumptions about how the score reflects how well that player played. For example, if in A Game of Thrones I make a decision that gives another player their seventh castle, the fact I have the second-most castles should probably not be a consolation.

Similarly, some games allow 'long play' strategies in which case they may have won had some other circumstance not ended the game early. The fact that even though they might have one given more time still doesn't mean they win, also seems to imply they shouldn't receive some kind victory for second.

Does doing so affect gameplay?

It seems that it probably does. From my experience, it definitely seems to effect gameplay in Settlers of Catan. When two players are behind a leader, it benefits them embargo the leader and work for mutual benefit. With a second place, they have to weigh up how much second place means to them against the possibility of beating the first player, leading to less cooperation.

Another situation is where a player will cause the game to end voluntarily when they are not winning, perhaps to preserve a second place. If there is no second place, then there is definitely no incentive to do this from a gameplay perspective.

Another scenario may be that given a choice between a high probablility at moving from third to second highest score before game end, and a low probability of moving from third to first highest score, the player needs to make a choice if there is a second place, else it's not a decision at all. Players should always take the options that might give them 1st.

Semi-cooperative games seem to make the issue even more stark; in those games if the group loses, then a second place makes little sense.

Has anyone written any articles on this subject?

What do most gaming groups choose for resolving this?

• I suggest reading en.wikipedia.org/wiki/Game_theory before I elaborate an asnwer when I have time. From what I read from your question, this is how computers play. en.wikipedia.org/wiki/Minimax is a basic strategy to win games with absolute knowledge.
– Kii
Jun 30, 2016 at 10:56
• In Catan tournaments I've seen, players' scores are converted into some tournament ranking, so it definitely matters. But on yucata.de online play (for all their games, which does not include Catan), points are only awarded on the basis of win/lose. So there clearly isn't a generally accepted interpretation. Jun 30, 2016 at 13:40
• This seems too broad and also pretty opinion based ("what do most gaming groups do?") - maybe narrow it down to e.g. 'how to determine positions after the winner for game X' - if you are interested in that... Jun 30, 2016 at 17:14
• Note that there are other options to determine players' ranking rather than score only - e.g. 'who is likely to reach the winning condition next'. Consider a Catan game for example, where player A has 10 points, player B has 8 points and is 2 resources short from building a city and a settlement (both of which he can generate from his territories), and Player C has 9 points but has just exhausted his resources in order to reach that state - it may be argued that B should be ranked 2nd, not C. In Essen 2014, players of Istanbul continued to play to determine ranking after first player has won. Jun 30, 2016 at 17:27
• @G0BLiN I guess my question is in the title. In a non-tournament environment is it correct to infer positions beyond first in a game that ranks players. I have been following the kickstarter for the new Mistborn game which can basically reverse the victory condition at game end (lowest score wins). This seems to muddle matters for me even more. Jul 1, 2016 at 0:45

Tl;dr: Players' meta-goals do affect gameplay. And unless all players have the same goals, the entire concept of "best player" isn't well defined, let alone "second best", etc.

First a bit of game-theory framework. Let's say each player has a "utility function" which takes as input the game end-state and gives as output how much "utility" they get - an abstract stand-in for happiness, bragging rights, etc. As an example, you might assign yourself 10/1/0 utils for getting 1st/2nd/3rd highest score in the game. A player's ultimate goal is solely to maximize their own expected utility, regardless of how much their opponents get. (Most utility functions in most games, however, are set up so that maximizing your own utility also minimizes your opponents' - e.g. if all players are using that example function, then to maximize yours you have to get more in-game points than everyone else, which minimizes theirs.)

Your choice of utility function absolutely affects gameplay. You give a bunch of accurate examples of this, so I won't go into it.

Now to the philosophy.

There isn't one "correct" utility function. Some games may have been made with one in mind - as you mention, some rules emphasize the winner while others tell you how to rank the players. But if one player is almost as happy with 2nd as 1st and another only really celebrates when they win outright, neither is likely to convince the other that they're wrong. Not only that but in most situations there's no need to - these two can happily play at the same table, each pursuing their slightly different goals. (Though as a sidenote, if these differences in goals remain hidden then some moves may appear irrational. This can lead to hard feelings, so it can be a good idea to discuss where you stand on this spectrum before you play.)

However, players who are playing with different functions are effectively playing (perhaps only slightly) different games. After all, since function affects gameplay, what is objectively a good move for one player might be a bad move for another! Thus if your goal in "inferring positions beyond the winner" is to determine how good the players are at the game relative to each other, well you can't even be certain about inferring that the winner is the best, let alone the other places!

As a concrete example, let's say you have a 3-player game in which Alice is average - say she can get 1st/2nd/3rd place 30%/40%/30% of the time. Bob starts out objectively better - he can get 1st/2nd/3rd place 35%/60%/5%. Then Alice decides she hates getting second and from now on makes a desperation push for 1st every single time, which fails more often than not, so her new values are 40%/0%/60%. I will now tell two stories. In the first story, Bob knows how to do the desperation move but he's happy with second place so he doesn't think it's worth it. Thus Alice wins more often than Bob (40% vs 35%), but Bob is still the better player. In the second story, there's a massive cash prize awarded to first place, and Bob can't figure out how to pull off that desperation move himself - he watches in frustration as Alice walks away with the prize 40% of the time. In this story, the percentages are exactly the same but this time Alice is clearly the better player.

Thus, unless you have a way of forcing everyone to use the same utility function, you might be unable to determine any rank - even best player - since the entire concept of best player isn't well defined. Luckily, this isn't usually a big problem since the people who really want to rank the players often do have a way of forcing everyone to use (roughly) the same utility function: those people are tournament hosts and that way is prizes. So when you're playing for fun you're welcome to your own concept of victory, and when you're in a tournament, well the prize for 1st in the second story above is not only the reward for being the best but also the reason we can define who's the best.

• "A player's ultimate goal is solely to maximize their own expected utility, regardless of how much their opponents get." That's one big assumption! A very interesting discussion though. Jul 1, 2016 at 9:53
• The utility framework is relatively assumption-free if we expand the domain of the functions. If a player cares about playing in a civil atmosphere, they can get -1 util for each angry word uttered. If a player's goal is that everyone else enjoy themselves, they could decide their utility function returns the sum of everyone else's, and now they just have to maximize their own. The cool part of the analysis is that even if we do restrict all players to only care about what place they get, the slightly different ways they can care are enough to make a definitive external ranking impossible. Jul 1, 2016 at 16:14
• @user2390246 - and it's sometimes a false assumption. I remember at one point my uni friends introduced The Prime Directive: "p*ss Andy off!". Their utility was most definitely affected by how many points I got! Jul 20, 2016 at 13:22
• Great point about incompatibility of utility functions - games can become un-fun when players have wildly different utility functions, as they're hardly even playing the same game. I've been on the receiving end of a scorched-earth Risk revenge campaign, where other players weren't actually trying to win, but just make sure that I didn't. It can be difficult to play against someone deriving utility from something orthogonal to the game rules. Feb 6 at 14:19

In a game where there is a scoring function, one can always rank players beyond the winner into 2nd, 3rd, etc. The real question is whether positions beyond first have any value. That question depends on many things, including player psychology, game design, and external incentives. Here are some examples of ways that this varies between games:

## Example 1: Poker

Poker is arguably a game of pure external incentives. You play with money, for money. The obvious scoring function is how much money players have at the end of the game, using how long players were in the game as the tiebreaker for eliminated players. One of the significant variants of poker is how cash is dolled out in the end. In one variant, known as a "cash game", players can leave whenever they want, taking with them however much money they have at that point. In anther variant often referred to as "tournament style", players play an elimination game until only one or a few players remain and the total money put in is given exclusively to the last player or the last few players.

Which variant is used causes significant difference in game-play. In cash games, the focus is on having an overall profitable strategy. Meanwhile, in tournament style, it is much more important to accumulate a drastic chip advantage and eliminate players, and there is usually a 1-on-1 showdown at the end (called "heads up") which plays very differently than poker at a larger table. Alternatively, if there is a big enough prize for second or third in a tournament style game, playing very conservatively and just trying to survive can be a very good strategy, despite it not doing very well in a tournament style game that only gives a a prize for first. While the mechanics of the game are the same between these variants, the strategies differ wildly.

## Example 2: Diplomacy

Like Poker, Diplomacy is a game with both an obvious scoring function and player elimination. Players control supply centers, each of which supports one military unit. A player wins by controlling 18 supply centers and a player with no supply centers is eliminated. Also of importance is that Diplomacy is a notoriously long game, often taking about 8 hours with the full 7 players if played to completion.

The rules of diplomacy describe the game as only one player wins, but in casual settings, players often form alliances and decide to declare their alliance as winning the game rather than playing the game out to its ultimate conclusion. In tournament Diplomacy, there is a provision for all players remaining in the game to declare joint victory, in which case all remaining players get a number of points equal to a fraction of the players remaining (so three players remaining in the game would each get 1/3 of a win). In a casual setting however, it is often hard for players to emotionally differentiate between 1/3 victory and 1 victory, especially given playing to full resolution can take another several hours. Whether or not players consider a joint victory to be a true victory causes massive differences in the integrity of alliances, since playing to true victory requires breaking alliances at some point. This causes huge differences in the outcomes of casual diplomacy games.

Another important factor in diplomacy is that if players have set aside a particular amount of time to play, getting eliminated early can be of significant dis-utility because then they are no longer part of the social event of the game. This can cause significant differences in game play, including players willing to make disadvantageous alliances simply to keep themselves alive. I played one game of diplomacy in which one player made a permanent alliance with me claiming he "just wanted to come in second"; it turned out to be true, and ended up wildly unbalancing the game in my favor.

## Example 3: Archipelago

Archipelago is an unusual game in that it is semi-cooperative. In most cases, a game ends with all players winning or losing together, but then there is also a points system used to define the "grand winner" in the case that all players win. Having played Archipelago, I have difficulty emotionally differentiating between "losing" and "winning but not coming in first". The game explicitly defines a "second place" that most players get, but it still kind of feels like losing. I care much more how close I came (points-wise) to being the grand winner in Archipelago than about the fact that I nominally won since how many points I have is an indicator of my performance. I believe the game would be just as good if it was defined as "one player wins or everyone loses" instead of defining grades of victory.

## Example 4: Games with Kingmaking

There is a fairly lengthy discussion on kingmaking here. The relevant aspect of kingmaking here is that players not in the running to win become rogue agents since they are no longer motivated by the stated goal of the game. This highlights that players usually are trying to achieve the stated goal of the game. Players then have secondary goals, which may include implicit standing at the end of the game, but can also include getting revenge on a player that screwed them, helping a person they like better, or doing something that amuses them. Kingmaking is one of the results of the enormous complexity underlying players' actions in games. The gist of it is that players' motivations are far more complex than a univariate utility function.

## Asside: Tournament Match Fixing

There is lots of external discussion on the topic. The relevant aspect of match fixing here is that players can be willing to lose games given proper external incentives. Consequently, tournaments often have very severe penalties against it to preserve the intended incentives of the game.

## Using Score for Future Games

The implicit places beyond first are a good approximation for the strength of your strategy, assuming that the outcome of a strategy is normally distributed. If you "come in last" in a game, it could be a result of bad luck or it could be a result of bad game-play, but either way usually requires a very different post-game analysis than if you "came in second". Coming in last often requires rethinking your entire approach, while coming in second may just be about tweaking the finer points. But even here, we care more about margin of victory than we do about ranks. If you "come in second" in a game of Catan, with the final scores being 10, 9, 6, and 3, you did much better than if you "come in second" with the final scores being 10, 5, 4, and 3.

Of course, this becomes less useful if the outcome of the game given a strategy is not normally distributed. Taking a single incredibly risky move that causes you to win a lot or lose a lot all at once (such as picking up an enormous pile of the discard in Rummy 500, keeping a very difficult set of routes in Ticket to Ride, or trying to shoot the moon in Hearts) will usually cause you to come in first or last depending on whether or not it succeeds. These moves are typically only good if coming in first (i.e. "winning) is all that matters, since if they fail, they will be very detrimental to your points. Taking these moves also makes your final points value a poor indicator of the performance of your strategy since the average is a worse approximation of the strategy's chances of success. For example, a strategy that scores 100 points 75% of the time and 0 points the remaining 25% will win far more games than a strategy that reliably scores 90 points, despite having a lower average score.

## Conclusion

Players have very complex utility functions when playing games. Some players try to win, no matter the cost. Some players try to do as well as possible relative to other players, and care about "coming in second" even if the game doesn't say that it counts for anything. Some players care about doing well relative to their past performance so that they can improve at the game. Some players will use the scores of the players as a tool to analyze the strategies employed by each. Some players play for motivations unrelated to the stated purpose of the game, such as to do something amusing or to achieve some external social goal. There is lots of overlap between these categories, and lots of things (like prizes) that can change players' utility when playing a game. I believe that implicit ranking is real and is always a factor in playing a game, but how much of a factor is entirely situational.

These answers seem to me to conflate two slightly different goals because of English usage considerations. Winner and "Best player" are not necessarily the same, In my Puerto Rico circle there is one player who is decidedly the "best", winning most of the time. When he loses he doesn't lose the assessment of being the "best" (most skillful) player, even if he loses this one game. (Though we all cheerfully assure him he has lost the gloss of "best".) So winning is a simple matter of scoring (as are 2nd, 3rd, etc.), and scoring one (or even several) game does not imply "best" player - especially given that most popular board games have some elements of randomness in order to keep them interesting.

• Randomness is orthogonal - I'm sure OP understands that it can prevent the best player from winning in a single game. I think the point of the question is whether "2nd most points" and "2nd best player" can be conflated ignoring randomness. In a 2-player game, they can - usually both players want to win, and so (over the long run) the player who wins more often is the better player. The point of my answer is that with 3 or more players this stops working - even if you play enough to overcome randomness, you can't ignore the differing values that people place on coming in 2nd. Jul 21, 2016 at 2:44

It depends on what the goal is.

Games like Scrabble or Yahtzee have scores on a fairly wide "continuum." So in games like these, it may be valid to use scores to measure "degrees of goodness."

Games like Catan have scores on a narrower continuum (7 or 8, up to 10). Such games are more "binary." You either achieve, or don't achieve the victory condition.

In the game of "Thrones," the basic victory condition is seven strongholds, but if no one achieves that, then smaller numbers are used to "rank," and hence determine the winner. (But that is a second choice.)

Like Catan, Thrones is a "first past the post game" (seven for Thrones, ten for Catan. In Scrabble or Yahtzee, there is no "post," only relative scores.