IMP tactics

[Phil]

There's a psychological element here I think.

 

If I go to the tables and I win $50, or lose $100, its no big deal either way.

 

If I win $500, its much more fun. Does the affect my strategy? Of course. I will tend to go for the bigger win, even though my payoff over the long term may be less, which is debatable anyway.

 

At bridge, no one (including you) cares about the up you ground out, much less the one that you sacrificed your contract to get. However, if you throw away a cold contract, then it will follow you around like a dark cloud.

 

The odds for overtricks would have to be MUCH better for me to ever put a contract at risk, and according to my gnomies, its pretty damn close anyway.

[jdonn]

I think there is logic that dictates that in longer knockout matches you should go for overtricks more aggressively than in short matches.

 

The other side of this argument is that in a short match there are fewer chances for swings and thus over and under tricks take on more importance. To take this idea to the extreme, imagine a one-board playoff: now the 1-imp can mean winning or losing. At the start of a 64-board match, that if one team had been given a 1-imp handicap it would be fairly meaningless.

Of course you can take it to both extremes. In a one trillion board match, if you don't take the maximum expected value then you are just playing badly. Interesting how the same conclusion can be reached at either end.

[JLOL]

I think a correction to the probabilities is in order. The chance that 1 imp will give you an extra VP is less than 1. I don't know the exact probability, because there are scores for which the probability is zero (such as us already leading or behind by 60 imps), but let's say it's somewhere around 1/3, because of the IMP bands around the VP scale. Now the 12 imp loss will affect our VP score at a much higher percentage (not 1, because we might already be losing by 60 or winning by enough that a 12 imp loss still means no VP swing). So when we make the calculations not in terms of an expected IMP swing, but rather an expected VP swing, we see that the 1 imp gain is only worth say 0.3 VPs, whereas the expected VP loss is worth probably something around 5 VPs. So now you need to have odds of 5.0:0.3 or around 94%. These figures make it very close and since I have approximations above, could not be used to make the decision in this case. However, what I'm saying is that we should be doing VP expectations, not IMP expectations.

Matt, I think 5 VPs for 12 imps is too high an average. If you were already up 2 big swings or down a big swing (or down 2 big swings) it probably will only cost 3 VPs.

 

I have no idea how to calculate how many VPs 12 imps will usually translate into vs how many VPs 1 imp will usually translate into, but my experience suggests that winning 1 imp is probably even more valuable in VP20 than in some knockout match. Hopefully Richard or someone can do a simulation to show this relationship. Probably it is pretty close either way to the imp odds.

 

Pretty much the only thing that I have agreed with Mikeh about has been that sometimes your upside is not even an imp because something weird has happened at the other table.

 

I think it is complete nonsense that "psychologically you cannot afford to go for it" because "your teammates will be mad, but they won't be mad if you lose 1 on an overtrick." It also should be complete nonsense that you will feel so bad if you go down that the upside isn't worth it. Your decisions should always be based on doing what is right, and making the best bridge decision, if your teammates or partner or even your own mind are hindering you from making what you think is the best bridge decision because of these extraneous irrational things then that is too bad, they are bad teammates. And if you personally make a good long term decision you are accepting the risk, so it is dumb to be upset when that risk becomes a reality.

 

It is also fallacious to say that "bridge tournaments are rarely decided by 1 VP, they are decided by the big swings." Well what if you win 5 VPs this way throughout the course of the tournament? Bridge tournaments are won by making good long term decisions in every possible situation. If the math shows that you are making a good decision then you will be winning more bridge tournaments by making it. It is just that simple. The only way this becomes untrue is if you are some kind of prohibitive favorite to win the event because you're so much better than everyone that you can afford to sacrifice some equity in these spots to avoid disaster. These situations rarely, if ever, come up.

 

I find it really remarkable that the solution to the problem of irrational thinking and risk aversion and being psychologically weak is to make worse plays rather than to try and work on this destructive thought process.

 

So Frances, to answer your question if those numbers were correct then yes I think the person played the right way. He even overlooked the chance of making 2 overtricks when someone has JTx of diamonds ;)

 

On the actual hand I think his play was definitely quite poor however because the clubs are much more likely to be 40 than the a priori percentages suggest given that they were bidding with nothing. This seems like a poor bridge decision.

[JLOL]

The same principle can be applied to defense - if there is a play that will defeat the contract 10% of the time but give away an overtrick 90% of the time do you make the play?

Yes, especially defending 1N at imps people give up overtricks as both defender and declarer way too often, sometimes even 2 overtricks. It is pretty silly to let them make 150 instead of 90 hoping for some obscure layout.

[helene_t]

However, what I'm saying is that we should be doing VP expectations, not IMP expectations.

The IMP->VP curve is sigmoid-like with a saddle point at zero, so if we are at tie or leading, the expected VP gain from a 1 IMP gain will be more than 1/10 of a 10 IMP gain, while the opposite will be true for losses, while if we are trailing it's the other way round. Similar considerations apply to the rawscore->IMP conversion. In both cases, of course the discreteness of the scores make it go in the opposite direction in many cases.

 

I think it's pretty obvious from the above that it isn't worthwhile to think about VPs. Just play bridge.

 

As for the psychology thing I agree 100% with Justin.

[Finch]

We had another hand a while ago with the complementary problem. I can't quite remember the hand, but in concept it went like this:

 

You are playing in a vulnerable 3NT contract at imps (not VPs this time, it was in the trials I think). You have a choice of two lines for your contract:

 

Line 1 is definitely the percentage line, and makes let's say 60% of the time

Line 2 is a worse line, and makes roughly 50% of the time

 

However, if line 1 fails you are going lots and lots off - possibly 4 or 5 off on a bad day. If Line 2 fails you are 1 or 2 off.

 

The discussion got very complex, because it was hugely difficult to work out the exact odds of the various lines, and the relative percentage chances of the number of undertricks. Certainly impossible at the table. But what happened was that:

 

Nick Sandqvist for the opposition took the percentage line and made the contract. His reasoning (we asked) was 'I played to make'

 

Team-mate took the other line and went off. His reasoning was that his imp expectation was higher by doing so. I think he was probably mathematically right, but it's interesting that the opposing player didn't think on those lines.

[JLOL]

Frances, I have seen hands like that and also think it is right to make the play that doesn't give you the best play to make to save a few vul undertricks when you're going down if the lines aren't too far apart. These situations are rare, and in my experience most very good players will just make the best line to make and say something like Nick Sandqvist said.

 

Honestly most great players are not great at thinking about the math of the game even in the abstract, they don't know most suit combinations (or even how to figure them out in a timely manner at the table), and probably have generally flawed thought processes about areas like these.

 

The truth is that these situations are marginal at best anyways, and saving a very few % here or there in rare spots or adding to your expected imp total by a very small amount in these overtrick/less undertrick spots which are also pretty rare doesn't end up is not that important compared to other areas where these great players will kill everyone else. I do think it's a very common leak though.

[BjornH]

Is not being in the right "state of mind" an issue here?

 

That is, if you focus solely on beating the contract or making the contract instead of overtricks/undertricks, you save energy that you (presumably) can use to make more/beat more contracts?

 

And that will be more rewarding in the long run (for most of us, anyway)?

 

Bjørn

[Trinidad]

Play Swiss teams, 7-board matches, IMPs converted to VPs on a 20-0 scale.

Your two hands are

 

Dealer: South Vul: Both Scoring: IMP

♠ 52 ♥ K1052 ♦ AK96 ♣ A32 ♠ A73 ♥ Q64 ♦ Q ♣ KQ9864

 

At both tables the auction starts 1C from South, a 1S overcall from West, double from North and 2S from East. The auctions then diverge.

 

At one table, your team plays in 5C on a spade lead. The play is mildly interesting, but you emerge with 11 tricks for +600.

 

At the other table, the opposing team play in 3NT on a spade lead. They go off, because clubs are 4-0 onside.  Declarer looks slightly embarrassed about going off in a makeable contract (B/I problem: what is the best technical line for 9 tricks?) but then defends himself by saying:

 

Yes, I could have won the spade lead, crossed to the ace of clubs and made 9 tricks by overtaking the queen of diamonds.  This picks up the 4-0 club break.  However, all the rest of the time I've given up an overtrick.  If they are +600 at the other table, then I've cost myself 1 imp 93% of the time (given that spades are 5-3) but avoided losing 12-13 imps 7% of the time. So I took the right line.

 

You could argue that his odds are slightly out, because the opponents have bid to the 2-level, vulnerable, on a combined 13-count with an 8-card fit so are more likely to have some shape.  But suppose he's right about the percentages.

 

Did he take the right line?

The part "Declarer looks slightly embarrassed" in combination with "but then" suggests to me that declarer came up with this after the hand was over. To me that makes it only of academic interest what the right line was (and academic interest can be very interesting).

 

But in real life, the key thing seems to be that declarer hadn't thought about the right line while he was playing the hand and that he came up with a justification after the fact.

 

Rik

[mikeh]

I think that there is a fallacy in jlol's reasoning.

 

We do not play infinitely long matches. I have played quite a few 64 board matches and several longer ones...up to 128 boards.

 

It is my experience, which I admit has not been based on taking notes, that the opportunities to make a mathematically appropriate overtrick play arise infrequently. Certainly, in a match of 7 boards, I doubt that the opportunity arises, on average more than once (if that).

 

If we are wrong, and lose 10 imps....we cannot regain that loss no matter how frequently the opportunity arises in that match, because there are never going to be more than 6 chances to recover an imp at a time, and usually no such opportunities.

 

It is only as the length of the match increases that we can rationalize losing 10 imps on, say, a 11-1 chance because this approach will, in the long run, result in a net gain. A round-robin scored in vps approaches that situation, if it is long enough. But a long event such as a multi-day swiss or a series of knockout matches doesn't. If you get knocked out due to losing 10 imps, you don't get to keep playing for the extra boards needed to get those imps back one at a time. And in the swiss, you may fall out of touch with the leaders, and picking up a lot of 1 imp swings lower in the pack won't get you back to the top.

 

Does this mean never risking a contract? Of course not.. but it does mean that one had better be damn sure of what one is doing... and consider all of the implications, rather than the simplistic and erroneous rationalization described by Frances.

 

Where the psychological problem arises is when a player gains a reputation for risking too many contracts. I know several players who analyze hands far better than I do, and who are far more successful at mps than I am, but I would never play imps with them, and they never seem to get on strong teams, because they play mps at imps... too many 'lose 10' trying for 'win 1', amongst other sins... and they will always be able to make the (simplistic) justification based on the odds of that hand.

 

Here's another way of looking at it.

 

Assume that in a long match, the situation arises 6 times.. where the odds seem to be 11-1 or 12-1 in favour of the overtrick. I don't remember enough math to work out the probability that you will lose one of these plays if you go for every one of them, but I think that it will be significant.. if it happens, then even if you get every other one right, you can't make up the loss of 10 imps when the play fails... but if the match were infinite, you could. Jlol's analysis would then be valid.

 

In the meantime, the 4 or 5 or 6 imps you pick up by 4 or 5 or 6 gambles are, usually, on the level of noise in the totality of the imps scored.. a typical match sees 2-4 imps being scored per board.

[helene_t]

Mike, you are making it way too complicated. Maximizing the expected IMPs (or even cruder: maximizing the expected raw score) is fine.

 

Take you example with 6 opportunities during the match to win 1 IMP by a 11/12 probability and losing 10 IMP with a 1/12 probability or such. If you go for the overtrick on all 6 you will probably either win 6, or you will lose 10 and win 5, that is a 5 IMP loss in total. Of course if you were to win by two IMPs by playing safe, going for the overtrick will now either make you win by 7 or lose by 4, but on the other hand, if you were going to lose by 2 IMPs otherwise your only chance is to go for the overtricks. Oh in fact you should go for the overtrick 3 times and play safe on the other 3 but you surely see why such a strategy is silly. There are too many unknowns: you don't know the state of the match, you don't know what they are doing at the other table.

 

Besides, it was Swiss teams, so you objective is not to win the match but to win as many VPs as possible. It is impossible to know how many IMPs you need to win to improve your VPs. It is possible that 10 IMPs will win two VPs while 1 IMP will win zero, in which case you should always play safe. It is also possible that 10 IMP will win 2 VPs while one IMP will win 1 VP, in which case you should go for the overtrick as long as the probability of making the contract is still better than 2/3. Etc.

[Stephen Tu]

Well, I think the main thing is that one must include mikeh's point of the chance the opponents landing in some stupid contract, so that you don't count phantom overtrick IMPs when it didn't matter and the IMP doesn't really exist. If you do include that in the math, I believe it's clear to go for the pos EV (VP/IMP depending on event) play, which would include going for overtricks when appropriate. I don't think length of match matters if we are talking about an early board, only state of match if known.

 

If we are wrong, and lose 10 imps....we cannot regain that loss no matter how frequently the opportunity arises in that match, because there are never going to be more than 6 chances to recover an imp at a time, and usually no such opportunities.

 

True enough, but the vast majority of the time we are not going to be wrong. Say we are playing a KO. Let's play this team 200 times. Let's say the KO is such length that this decision only comes up on average once. 95% we win an imp, 5% we lose 10. Now of course it's much more likely that 10 imps will swing a match from a loss to a win, than 1 imp. But since the one imp comes in so much more often, that's OK. Say 30% of the time we were within 9 on the other boards, so the 10 imps wins us 3 matches. Say we are tied only 2.1% of the time. That gives us 4 wins. 1 net extra win vs. playing safe.

 

Assume that in a long match, the situation arises 6 times.. where the odds seem to be 11-1 or 12-1 in favour of the overtrick. I don't remember enough math to work out the probability that you will lose one of these plays if you go for every one of them, but I think that it will be significant.

If it's 12-1 in favor, and you do it 6 times, then you are (12/13)^6 of winning them all, so 61.86% you are up 6 imps. 30.93% you are down 5, 6.44% down 16.

[JLOL]

1) Regarding the argument of the VP to IMP relation I will just echo helene, assuming no state of the match considerations if you are playing to maximize your imp expectation you are very likely maximizing your VP expectation. Han has volunteered to do some calculations on this, but I will just say that I definitely agree with Helene and not Mike about this.

 

2) Regarding "you will not reach the long run in a match so don't take the risk," that just makes no sense. That's like saying don't bid a 40 % vul game in z short imp match because you may only bid 1 vul game, and you will be going down more often than you make! Of course you still do the right thing over and over again even though you may get unlucky in a short match and lose; you are still increasing your chance to win. You are going to win 1 imp so often that even if you only get this chance once in your entire life it's a good one to take. The only way this is not true is if the 1 imp is worthless which just goes back to argument 1.

 

3) Regarding never playing on a team with someone who will risk their contract for an overtrick, the people you describe seem to not be making the right assumptions about when it is right to go for it, or are the most unlucky people ever. They are clearly making some errors in their calculations if they are going down with any frequency in cold games, for instance on this hand given the bidding I think it would be definitely wrong to go for it. If your point is that most people who do this stuff do it in the wrong situations because they are not making good bridge assumptions then fine, but personally I would never want to play on a team with someone who blew an overtrick for no good reason.

 

Honestly I don't understand why instead of trying to make the right play we are ever thinking about things like "ZOMG MY TEAMMATES MIGHT BE MAD!" or "OMG I WOULD FEEL SAD IF I WENT DOWN." It seems to be part of your thought process routinely what your teammates or partner will think, and what you would think if your teammate did this. In my opinion this is not a great way to play bridge.

[mikeh]

My suspicion is that in practice we would probably agree on 99% or more of the hands on which the issue arose.. I never said that one should never play for the overtrick.. I said that one should only do so in some circumstances... a tight match towards the end of the event being one I specified, but I also said if one was damn sure that it was right.

 

Frances posted a hand on which the declarer was able to ex post facto argue that the odds were 93-7 in his favour, or about 13-1 and break-even would be 10-1. You, and others including me, think that the auction should have impacted his assessment of the odds. I, and others including you, think that he should also have considered the non-trivial odds that the 1 imp he was playing for didn't exist due to other table issues.

 

So, I think we are agreed, even leaving aside the VP issues, that we need some vigorish on the exact on the hand odds... even after accounting for inferences from the auction, we need a fudge factor for the risk that 1 imp is simply unavailable due to other table action... to ignore that is silly, but how much weight to give it is going to depend on how easy the hand looks (how predictable the bidding) and some knowledge of the opps and your teammates (who might, on some hands, be jamming the auction). So no hard and fast rule can be stipulated. On some hands, the vigorish may be zero, on others (such as where we have survived a barrage and landed on our feet in a difficult-to-reach contract) it may be overwhelming. In the given hand, I think there should be a modest amount... certainly enough that the odds for the play, in the context of the two hands with no opposing auction, should be better than 13-1 to justify the risk.

 

As for my repeated references to partners and teammates.... I have never been concerned with reactions to sensible choices that failed.. but I am concerned about reactions to poor choices... errors.. that cost.. and on this hand, as posted, in my view, playing for the overtrick was an error. if it was caused by fatigue or temporary loss of focus.. we all suffer from that. If it was caused by not appreciating the difference between imps and mps, then that would worry me to the point that if it happened several times, I'd make a note that (absent other very positive aspects of his game) this player is not a strong teammate. I don't play pro.. I play with friends, and when I want to do well in an event, I try to play with strong teammates who are also friends... bearing in mind that I don't travel so I'm not looking for teammates for the Vanderbilt, etc :(

[JLOL]

Agree with your last post.

 

Also, funny that you mention playing pro, you probably would be even more conservative in going for overtricks if you did since you'd have job security issues to worry about lol.

[orlam]

Wow interesting thread, but so much nonsensical math has been posted...

[gnasher]

a typical match sees 2-4 imps being scored per board.

Do you play a lot of junior bridge?

 

Edit: Sorry, maybe Mike's right: I've just looked at the scorecard for what I thought was a fairly unexciting match last night, and 74 IMPs seem to have changed hands over the last 16 boards.

[whereagles]

Just for the record, I once lost a 20 board international match by 20-10 solely on overtricks.

[jdonn]

a typical match sees 2-4 imps being scored per board.

Do you play a lot of junior bridge?

It seems he simply plays good bridge.

 

In the last Bermuda Bowl, these were the average imps scored per board in the knockout phase:

Quarterfinals: 3.6, 3.3, 3.9, 3.3

Semifinals: 4.1, 5.4

Finals: 4.4

The average of those (rounded) totals: 4.00

 

Venice Cup:

Quarterfinals: 3.4, 3.6, 4.3, 3.6

Semifinals: 4.3, 5.1

Finals: 4.1

The average of those (rounded) totals: 4.06

 

It looks like you were correct to suggest Mike was off on his estimate, but because it was far too low, not too high. Not 2-4 imps a board, more like 3.3-5.4 imps a board.

 

Contrary to popular prejudice, juniors are far more sane, as they range from 3.2-4.7 imps a board.

 

World Mind Sports Games under 28 event:

Quarterfinals: 4.7, 3.2, 3.6, 4.2

Semifinals: 3.7, 3.7

Finals: 3.6

The average of those (rounded) totals: 3.81

 

But those were the old juniors, I should be fair and examine those crazy young juniors instead!

 

Under 20 event:

Quarterfinals: 4.1, 3.5, 3.4, 3.8

Semifinals: 4.5, 3.7

Finals: 4.2

The average of those (rounded) totals: 3.89

 

Oh, never mind...

[Rossoneri]

Assume that in a long match, the situation arises 6 times.. where the odds seem to be 11-1 or 12-1 in favour of the overtrick. I don't remember enough math to work out the probability that you will lose one of these plays if you go for every one of them, but I think that it will be significant.. if it happens, then even if you get every other one right, you can't make up the loss of 10 imps when the play fails... but if the match were infinite, you could. Jlol's analysis would then be valid.

 

In the meantime, the 4 or 5 or 6 imps you pick up by 4 or 5 or 6 gambles are, usually, on the level of noise in the totality of the imps scored.. a typical match sees 2-4 imps being scored per board.

I remember a particular time more than 1 year ago when I was in camp doing duty one day and I thought of something along these lines.

 

My assumptions were:

1) You either make the overtrick, or go down

2) Assume this decision happens every board

3) This is an 8-board match

 

Given the fact that it was nearly 2 years since I had stopped doing maths at that point in time, I attempted to calculate this using a binomial model. My conclusion then was that the payoff of an imp wasn't worth it.

 

The paper on which I did my workings that day has since been lost, but after slightly more than one term of restarting school again, I am now still pretty sure my simplistic model calculations were correct.

[ceeb]

Simply on the question of IMP odds versus VP odds I've done some analysis. In summary, under typical assumptions there's very little difference but to my surprise VP does not encourage risk.

 

Why is that? Doesn't the VP scale (like the IMP scale) compress large swings? As Helene said, it's sigmoid isn't it? I always assumed so. But a close look shows this isn't a very good approximation.

 

Take for example the WBF 14-board scale. There are 5 imp differences (-2 to +2 imps difference) that correspond to a VP tie. And 5 imp differences (3-7 imps) that give one more VP. But the next bracket (8-10 imp) is smaller. The sequence of imp bracket sizes for winning 15-25 VP is (5 5 3 4 4 4 4 4 4 4 infinite). Hence the relation between imp and VP is better described as roughly linear but irregular, than concave upward.

 

As a concrete example I assume the WBF 14-board VP scale, and that the imp consideration is risking 10 imps to gain 1.

 

Hence in IMP terms it is break-even to try for the overtrick provided the chance of success is 10/11. A few comparisons of this number with the VP situation:

 

Typical example: you estimate the match state as variously anywhere from even to plus or minus 40 imps with a standard deviation of imp uncertainty (and this includes both uncertainty in guessing the match state as well as volatility of future boards) of 17 imp. Break-even to try for the overtrick is 10.003/11 chance of success when behind to as much as 10.2/11 when far ahead -- you should be microscopically more conservative at VP.

 

Extreme example: Suppose in the artificial extreme that your estimate of 10 imp lead has a standard deviation of only one imp. Then 7.3/11 chance is sufficient to justify taking the risk at VP.

[skjaeran]

Or our opps reach slam.... and make it... they score 1370 and we lose either 740 or 770.. again, the overtrick is irrelevant. Or they go down.. if we score 600 or 630.. no difference, but if we go down when we had 600 cold... we lose big.

740 converts to 12 IMPs, 770 to 13 IMPs....

But that's a minor issue. I agree with Mike. :)

[dburn]

Jeff Rubens, who is a more than competent enough bridge-playing mathematician to understand these things, has referred more than once to a "parabolic utility function" in respect of overtricks. By this I think he means that you should play for overtricks if the match is either "sufficiently short" or "sufficiently long" to make doing so worthwhile.

 

Obviously you should take a 51% line for an overtrick as opposed to a 100% line that will always make exactly if the match is one board long - but I wonder: how long does a match have to be so that you should play for "percentage overtricks" (that is, say, taking a 91% line for an overtrick that will gain 1 IMP or lose 10) at every available opportunity?

[pretzalz]

A psychological aspect that I don't think has been mentioned: I never feel more demoralized at the half then for it to seem like we've lost 1 or 2 on almost every board. Most overtricks aren't taking a chance; they are simply better technique or careless defense. I'd rather be down 20 with the dribs and drabs coming our way than down 10 with the dribs and drabs ebbing away from us.

[ceeb]

Jeff Rubens, who is a more than competent enough bridge-playing mathematician to understand these things, has referred more than once to a "parabolic utility function" in respect of overtricks. By this I think he means that you should play for overtricks if the match is either "sufficiently short" or "sufficiently long" to make doing so worthwhile.

 

Obviously you should take a 51% line for an overtrick as opposed to a 100% line that will always make exactly if the match is one board long - but I wonder: how long does a match have to be so that you should play for "percentage overtricks" (that is, say, taking a 91% line for an overtrick that will gain 1 IMP or lose 10) at every available opportunity?

I love a paradoxical result as much as anyone, but I don't see where such a result comes from. The behavior of the model that makes sense to me is not so intriguing. Rather --

 

In a 1-board match, I agree that a 51% chance merits playing for the 1 imp gain at the risk of a 10 imp loss.

 

However, as the number of boards increases, the necessary odds simply increase monotonically asymptotically to the imp odds. That is, the longer the match, the closer to 90.9090...% must be your chance of success before it's worth risking the contract for an overtrick. I don't see any increase then decline, or the opposite.

 

My model is as follows. The object is to end the match plus imps, no matter the margin. If the IMPs finish even, assume a coin-flip playoff. Other than the present board, assume the teams are even with a normal probability distribution assigned to the various IMP margins -- i.e. the largest probability for a tie, slightly less for + or - one imp, etc. The spread of this distribution -- i.e. the standard deviation -- is identically 0 for the zero other boards of a one-board match, and the longer the match the broader the spread.

 

It doesn't require many boards before the threshold percentage to justify risking the contract is within a trivial margin of the imp odds. I.e. at a standard deviation of 10 imps, you need an 89.5% chance to justify gambling for an overtrick.

 

Charles Brenner