In Spades, what is the meaning of a high/low sum of bids?

Question

In Spades, when playing with 4 competent players, the sum of the four bids on most rounds is about 11. However some rounds have a very high or low sum of bids, mostly due to rare hands distribution.

1. What can a player deduce about the other hands when facing a high sum of bids of 12-13?
2. What can a player deduce about the other hands when facing a low sum of bids of 8-9?

to simplify the question assume no Nil bids were made.

Edit: David Robie gave a very good answer for the meaning of individual high/bid bid meaning, however I am looking for the meaning of the sum-of-bids.
For example, can a player assume in a round with high bid that the probability for trumping her side-suit high cards is higher?

Answer 1

A low/high sum of bids can mean anything. The sum of 4,4,4,3 is the same as 1,1,1,12 (exaggeration), but those two mean completely different things. Instead of looking at the sum of the bids, you get more information from looking at the individual bids.

It's safe to say that an Ace or king of any suit is likely to take the trick it is played in. As there are 8 of these a player will typically get at least 2, and bid on them. Players receive 13 cards, so your average is 3 cards of each suit, plus one of any suit. On a true average deal, this means that at least one opponent has 4 cards of one non-spade suit, and you will typically exhaust your supply of that suit before them, allowing you to play a spade if they lead with that suit. Any spade suit card has a higher value than any non-spade card, so you would win that trick if you play a higher spade than the other opponent. If you have 4 spades and 3 of the other suits, you are also likely to trump (play spade on a non-spade trick) a trick and take it. In either of these two scenarios, a player is likely to bid on at least 1 (if not 2) low-mid value spade card. Therefore, on average, a competent player might bid 3-4. Now let's look at higher and lower bids from an individual, and what that means.

A low bid could be caused by a number of things:

1. Having multiple cards in the 2-8 range means you are unlikely to take many tricks.
2. Having few to no spade cards means the other 3 players have them, and you are much less likely to take tricks by your own merit in the late-game.
3. Having many cards of one non-spade suit is a huge detriment. Not only do you yourself not have many spades for the late-game, but your opponents can start trumping you sooner, and if a suit is led with spades you might not be able to compete for that trick.

Now let's look at a high bid, which is basically the opposite:

1. Many cards in the high range increases your chances of taking a round.
2. Many spade cards means you can trump your opponents sooner/more regularly. If you have all 13 spades, you take every single trick in the round.
3. Few to no cards of a non-spade suit is a great hand, because you can remove low value non-spade cards early in the round, and trump high value non-spade cards played by opponents.
4. If you play by the rule that your first card played is the lowest club in your hand, and your lowest club is 8+, it's safe to bid on that trick.

Overall, a low/high sum of bids doesn't say anything about hands, because you don't know if there are outliers. A low/high individual bid can give you some insight as to the suit distribution and number value of cards held by that player. If you pay attention to someone's bid, and watch the first few tricks they play, you can get a good idea of what the rest of their hand is.

Answer 2

If an analog of Aumann's agreement theorem applied, then the bids would always add to 13. Why do they differ (let's say, to be more specific, that the bids are less than 13; the logic applies similarly to being more than 13, with the obvious modifications)?

One condition that may explain it is that players not be bidding perfectly: overall, players are underestimating the strength of their hands. Another reason is that players are not perfectly confident that other players are bidding perfectly: Bob may see Alice submitting a low bid, but not be sure whether that low bid is due to a bad hand, or due to the Alice underestimating the strength of her hand. So Bob will not adjust his bid as much as he would if he were sure that Alice's bid was due to her having a bad hand. Players also could lack confidence that other players are confident in other players' skill. Suppose Alice and then Bob both submit low bid, and Charlie is next to bid. If he were certain that Alice and Bob are perfect bidders, and that Bob knows that Alice is a perfect bidder, then he would conclude that Bob must have a very bad hand, to bid low even after learning that Alice thinks she has a low hand. But Charlie might think that Bob thinks that Alice might have underestimated the strength of her hand. If Charlie thinks that, then his estimation of Bob's hand's strength will be higher than if he were certain that Bob is certain that Alice is a perfect bidder. It's not necessary for Bob to be uncertain of Alice's skill, only that Charlie is uncertain whether Bob is uncertain of Alice's skill. And of course, if Debra bids next, and she is uncertain as to whether Charlie is uncertain of whether Bob is certain that Alice is a perfect bidder, that can cause Debra to underbid. Thus, a low total bid can occur if the players have hands that look individually bad (but aren't as bad as they look, because the other hands are also bad), and the players don't have perfect meta-knowledge.

The other way that Aumann's theorem fails to hold is that the bids are not iterative: Alice has no opportunity to adjust her bid upon learning that the other players think that their hands are bad. Any decent player, if they did have such an opportunity, would tend to adjust their bid upwards. The earlier the player bids, the less information they have from the other players' bid, and so the more likely their bids are the source of the total underbid.

Another factor is that the cost of overbidding, versus the cost of underbidding, are not generally the same, and players will err towards the direction with a lower cost. So a total underbid could mean that players are uncertain how many tricks they can take, and underbidding is less costly than overbidding.

So, overall, a low total bid suggests that the later players actually have bad hands, while the earlier players have hands that look bad individually, but aren't actually as bad as those players thought they were, because they were unable to take into account how bad the later players' hand are. However, low skill or low metaknowledge regarding the other players' skill can complicate this analysis, as can nonsymmetric cost functions.