[benlessard]

Assuming that you have 4 perfect robot players who know each other "programming" perfectly. So no matter what style they have their partner understand the style 100% and will bid accordingly, if you adjust yourself to fight the opponents style you bot partner will follow you and be 100% on the same wavelenght. I understand that IRL knowing partner style/tendency is much more important than the style/tendency itself but here we are assuming that the bots are in perfect harmony.

Bidding start

N-----E-----S------W
1Nt--(xy)--??

where xy is natural and lets say from 2C to 3S.

Can game theory teach us something here ?

1- There is many equilibirum "overcall". These are hands where overcalling or not will give you the same long term result against opponents (optimal ?) strategies. Make these hands a bit better for overcall and you should overcall with it, while not doing so will make you lose some EV. Make these hands less suitables and overcalling become a losing proposition.

2- the more benefits the overcaller can get (lead directing & stolen space) the more risk he is willing to take. Therefore the more space NS lose (not on a particular hand but on a whole sets of hands) the more they have to retaliate by having a higher % of doubled contracts.

3- Is there an optimal % of doubled contract that should be reached ? If there is one, being too low should penalize you and too high is not good either (either they make it too often or you agressive penalty aproach hindered your partscore battle). This % is going to be different for every different situation/overcall East make, (changing vul = different situation.

4- the lower the overcall, the more space NS have to compete, so a doubled contract is less likely to be desirable/needed.

5- Lets say that you are EW and feel that NS dont reached enough doubled contract, should you start to make weaker overcall ? (since the risk is reduced & assuming NS wont adapt to your strategy)

6- If NS tend to double too much Should EW thighten their overcall requirements ?

7- Is there many equivalent ways to reached the correct % of doubled contract ? Should a perfect penalty X = perfect DSIP/trap pass = take out X converted/trap pass

8- If you dont get your fair share of quota of doubling contract would getting to better partscore some of the times be a compensation (doubling them a fair amount of times is not an absolute ?)

Ive got some good questions and comments in reserve but im going to stop for now waiting to see others people comments.

[awm]

There can be multiple equilibria. They should have equal average score, but they don't need to have the same "doubled contract probability" so there isn't a unique value for that. Also, being doubled is not the only way for E/W to get a bad score in this auction, nor does being doubled necessarily result in a bad score!

The only conclusion I see is that if I am getting good scores when I overcall then I should overcall more, and vice versa.

[phil_20686]

awm, on 2012-March-22, 15:00, said:

The only conclusion I see is that if I am getting good scores when I overcall then I should overcall more, and vice versa.

This is wrong.

Firstly you must look at the average score of both the boards where you overcall, and those where you didn't overcall, but might have. So if you are getting good scores when you overcall, and poor scores when you don't, then you should overcall more. You do this until the extra overcalls are reducing the imp average of over calling more than they are increasing the imp average of not overcalling. If this is the case you have reached an optimal equilibria.

However, as pointed out, there are probably multiple equilibria, and there may well be several cross overs where the principle benefits of of each style get lost. In this case, there is a destructive and constructive case for overcalling, and there is probably an equilibria for each case. It may even be the case that for hands above a certain strength you do best by not overcalling and taking the vul undertricks.

A completely unscientific anecdotal evidence suggests to me that there are two principle styles for overcalling, the sound and the light, and that there is a gap between these, with fewer people having a minimum inbetween.

[phil_20686]

awm, on 2012-March-22, 15:00, said:

There can be multiple equilibria. They should have equal average score.

Also, as I understand it, an equilibria means that any person changing their style a little leads to a worse result. That is to say that "equilibrium" is a local condition, and therefore they do not have to reach the best equilibrium. And indeed this may be impossible to reach via incremental adjustments of style.

[dake50]

Start with a big survey of hands analyzed.
Tabulate penalty double success and competing double success.
That at least gets a reliable comparison.
Whether that can be translated into better theory is another question.

[xcurt]

This problem is harder than has been made out because usually we are maximizing not our expected score in points, but our expected score in IMPs or matchpoints. There are 4 more bots at the other table, and Condorcet cycles can arise. So unless you constrain the allowed strategies to one decision per table (now its just prisoners dilemma), or specify the other table strategy choices, or restrict yourself to rubber bridge, given the way you specified the problem you will wind up with a Nash equilibrium where the omniscient bots calculate strategies that are probability distributions over the set of choices. These are intesting problems, but they aren't going to get you to any ground truth about overcalling styles, which is how I understood the OPs question. These situations are also easier studied in simpler games like simplified forms of poker.

[awm]

When I said "getting good scores when I overcall" I thought it was obvious that this meant "relative to the scores I get when I don't overcall;" sorry if this wasn't clear to Phil_20686.

In general Nash equilibrium doesn't require a unique score and might be a local (only) optimum, but bridge is a zero-sum game so this doesn't apply. If there are multiple equilibria they will be equally good.