Nigel Merrigan wrote an article on backgammon races, which discusses some work on racing theory. The main addition is a formula for estimating the probability of winning a long race, called the Metric formula.
4.80
14 Dec 2011 - by Douglas Zare
Races are the simplest part of backgammon, but they still produce interesting checker plays. Let us continue to study racing to save the gammon. In this column we will use a database to find the optimal plays.
4.60
07 Nov 2011 - by Douglas Zare
Even the last few rolls of a backgammon race can have some interesting checker play decisions. In this column we will look at some of the decisions about filling gaps in your board and the effective pip count.
4.80
03 Oct 2011 - by Douglas Zare
Backgammon is too complicated to solve in the near future, but pieces of the game can be analyzed completely. If you have two checkers to race home, you have a few decisions. What should you try to accomplish?
4.40
01 Sep 2011 - by Douglas Zare
We understand backgammon races pretty well using tools such as the pip count and effective pip count. What about contact positions where each side has only one checker in play?
4.86
01 Aug 2011 - by Douglas Zare
In a past column, we looked at streaks in backgammon such as rolling doubles 5 times in a row. It is also interesting to look at clumps, or clusters of events. For example, in a recent game, I rolled 10 doubles in 24 rolls.
4.00
01 Jun 2011 - by Douglas Zare
Backgammon is a game of skill and luck, so there are times you will win even if your opponent outplayed you. You will also lose some matches against players who are playing poorly but rolling well.
4.63
03 May 2011 - by Douglas Zare
Backgammon match play is like a pentathlon from track and field. An athlete training for a pentathlon usually concentrates on one of the five events at a time, perhaps running one time, and then throwing a javelin another.
3.86
01 Apr 2011 - by Douglas Zare
A lot of match equity is at stake in each game after the Crawford game, and also on large cubes, but the backgammon literature says little about the intersection of these situations. This column fills that appropriate gap.
3.78
02 Mar 2011 - by Douglas Zare
Backgammon programs offer many statistics after analyzing a match. This feedback is valuable, but at times it is confusing since the statistics are complicated. In fact, some values reported by the bots have almost no meaning.
3.67
03 Feb 2011 - by Douglas Zare
Backgammon match play is tricky since the points do not have the same value. It gets particularly interesting when the doubling cube is on 2 or higher, and those decisions are often very far from decisions in a money session.
4.78
04 Jan 2011 - by Douglas Zare
When we play backgammon matches, we face a series of decisions at one match score, and then move on to the next score. It is an enlightening exercise to do the reverse.
4.50
02 Dec 2010 - by Douglas Zare
Backgammon involves weighing risks against each other. As humans, we don't have to decide how much each position is worth, only which play is best.
4.82
03 Nov 2010 - by Douglas Zare
We continue our study of backgammon after closing out the last checker. In this column, we will look at the races and contact positions which arise after the last checker enters, although this will have consequences even for the early bearoff.
4.91
04 Oct 2010 - by Douglas Zare
We continue our study of backgammon after closing out the last checker. In Part 1, we looked at the early bearoff. In Part 2, we will concentrate on positions where the last checker is still on the bar.
4.50
02 Sep 2010 - by Douglas Zare
One backgammon milestone I have yet to accomplish is winning in real play after closing out my opponent's 15th checker. My style of play does not lead to as many backgames and desperate ace point games which lead to hitting the last checker.
4.82
03 Aug 2010 - by Douglas Zare
Backgammon positions which are known to be close take/pass decisions are commonly studied. Knowing which positions are on the boundary can help you to make the right decision.
4.75
01 Jul 2010 - by Douglas Zare
What good does studying a single backgammon position do? Except for a few positions in the opening or endgame, or in a proposition, you are unlikely to hit that exact position again.
4.18
07 Jun 2010 - by Douglas Zare
Probability is important in backgammon. We make decisions under risk, and we have to interpret noisy evidence, such as playing out a position a single time when it would take a rollout with thousands of trials to be conclusive.
4.56
01 May 2010 - by Douglas Zare
A key idea in backgammon is equity, the average value of a position assuming perfect play. This is a weighted average over all of the possible sequences of rolls, giving the sequences appropriate weights assuming the dice are fair.
4.14
01 Apr 2010 - by Douglas Zare
If the dice are fair, then when you look back over your records, you should find that 1/6 of the dice are 6s, right? What else could be meant by fair dice? However, by the end of the column, I hope it won't be so obvious.
4.00
01 Mar 2010 - by Douglas Zare
Usually, when people play backgammon online, it is on a server which generates the dice. Despite the complaints about every single server, people trust the servers' dice. How can you tell if a server's dice are fair?
4.50
01 Feb 2010 - by Douglas Zare
In this column, we'll see a simple, computable racing formula due to Danny Kleinman. We'll use it to analyze a backgammon checker play decision which depends on correctly evaluating two desperate races.
4.19
01 Jan 2010 - by Douglas Zare
What can backgammon players learn from the financial crisis of 2008-2009? The same mental errors which led to the failures of so many banks and companies show up in chouettes and your private struggle against the bots.
4.53
01 Dec 2009 - by Douglas Zare
Backgammon programs are good at saying what the right play is, but they aren't good at explaining why. In this column, we'll look at some exercises for using bot feedback to understand why a decision is right so you can play better.
4.24
02 Nov 2009 - by Douglas Zare
In the short run, a game of skill and chance like backgammon might be viewed as a game of chess for $1 plus a coin flip for $10. In the long run it might be described as a game of chess for $1000 plus a coin flip for $100.
4.45
01 Oct 2009 - by Douglas Zare
Duplication is one of the most fundamental tactical ideas in backgammon. Understanding the basics of duplication might separate the intermediate player from the novice, but there are aspects of duplication worth studying by experts.
4.62
24 Aug 2009 - by Douglas Zare
Walter Trice just passed away. I'm still reeling from the news. He was my friend and collaborator, a role model, and an advisor.
3.67
01 Aug 2009 - by Douglas Zare
The international standard for backgammon matches is that they use the doubling cube with the Crawford Rule. However, you can occasionally find matches played without the doubling cube, or without the Crawford rule.
4.70
01 Jul 2009 - by Douglas Zare
Douglas Zare's encounter with the largest EMG checker play blunder he's ever seen reminds him of some advanced plays involving casual players in live backgammon clubs.
4.00
01 Jun 2009 - by Douglas Zare
This is the third and final part of our survey of responses to opening rolls according to Gnu Backgammon and Snowie, judged by long rollouts.
4.82
01 May 2009 - by Douglas Zare
Gnu Backgammon and Snowie are both very strong, but in about 15% of responses to the opening roll, they disagree with each other, or with rollouts. Let's continue our survey, using long Gnu Backgammon rollouts as the referee.
4.92
01 Apr 2009 - by Douglas Zare
Gnu Backgammon and Snowie are both extremely strong backgammon players. Some types of positions are better understood by one bot than the other. Which bot understands the opening better?
4.64
01 Mar 2009 - by Douglas Zare
Backgammon bots are usually great at judging plays, but occasionally they favor an insane play which no human player would seriously prefer. Rollouts often seem to confirm the insane play.
4.59
01 Feb 2009 - by Douglas Zare
Backgammon remains exciting to experienced players and comprehensible to beginners because most games have both easy and tough decisions. A few decisions per game are difficult even for experts, whether we realize they are difficult or not.
4.00
01 Jan 2009 - by Douglas Zare
Timing may be the most difficult of the important backgammon concepts to express in words. Timing in backgammon refers to whether you can maintain your assets long enough to use them.
4.33
01 Dec 2008 - by Douglas Zare
The effective pip count is particularly useful for understanding inefficient backgammon races. In this column, we will look at two epc formulas found by Walter Trice, for 0 or 1 straggler, and we will extend them to the case of 2 stragglers.
4.67
01 Nov 2008 - by Douglas Zare
Using any reasonable match equity table is better than not using anything, but now there are multiple tables in use. Woolsey has a commonly used MET, but Snowie uses its own MET, and GNU uses a different table by default. What happens when the tables conflict?
4.79
01 Oct 2008 - by Douglas Zare
When a backgammon game becomes an even race, you should aim for the 753 position, with 7 checkers on the 6 point, 5 on the 5 point, and 3 on the 4 point. In practice, we rarely achieve it. What should you do?
4.27
01 Aug 2008 - by Douglas Zare
Backgammon's doubling cube means you don't have to bear off all of your checkers to win. The virtual end zone tells you the game winning chances you need to plan to reach to win the game with the doubling cube.
3.60
01 Jul 2008 - by Douglas Zare
Some of the things we learn from studying backgammon give us an advantage elsewhere. Strong backgammon players have succeeded in many areas, from other games such as poker, to the financial markets.
4.73
01 Jun 2008 - by Douglas Zare
In this column, we will look at the theory of deciding between plays based on experience. This may help to see where our backgammon-playing experience is insufficient, and needs the most help from other learning methods.
4.38
01 May 2008 - by Douglas Zare
In backgammon, when 94% of your opponent's checkers get by you, you can still have a commanding advantage, or decent winning chances even with only one obstacle in front of your opponent's last straggler.
4.50
01 Apr 2008 - by Douglas Zare
The initiative of a backgammon position is the equity of the position when you are on roll minus the equity when your opponent is on roll. Studying the initiative is an exercise for understanding backgammon positions better.
3.77
01 Mar 2008 - by Douglas Zare
In a past column, we quantified the value from using the doubling cube, and applied that to understanding some take/pass decisions in money play. In this column, we'll continue to look at cube vigorish, but the application will be to a checker play decision in a backgammon match.
4.69
01 Feb 2008 - by Douglas Zare
Luck is an essential part of backgammon. Some players go to great lengths to tell you how unlucky they are and how lucky you are. In this column, we'll look at an exercise I use which has the side effect of being able to trap some of the excessive complainers in live games.
4.17
01 Jan 2008 - by Douglas Zare
Avoiding gammons and backgammons is not flashy, but proper technique in grim situations is part of playing well. At the end of the session, points gained from winning extra gammons are just as valuable as points saved from losing fewer gammons.
4.63
01 Dec 2007 - by Douglas Zare
Many backgammon clubs have a Grand Prix-style points race each year. Players receive points for placing well in the club's tournaments, and can never lose points. If you want to reward playing well, why award any points to second place?
3.50
01 Nov 2007 - by Douglas Zare
I've been planning to write about testing claims of cheating and unfair dice in backgammon. While I will discuss some of the foundations needed to analyze these claims fairly, this column is about the current cheating scandal involving Absolute Poker, one of the largest online poker sites.
4.59
01 Oct 2007 - by Douglas Zare
Where are the most serious backgammon errors committed? Are the most damaging errors well-practiced mistakes in the opening, misjudgments in the middle game, or poor technique in the endgame?
4.20
01 Sep 2007 - by Douglas Zare
The most serious errors in backgammon are not necessarily the largest. Just as more equity is given up on checker play errors than cube blunders, common errors may contribute more to your error rate than blunders in rare positions.
4.67
01 Aug 2007 - by Douglas Zare
In my June 2007 column, "The Essence of Backgammon," I remarked that the game of checkers is too complicated to be completely solved at present. Researchers at the University of Alberta recently announced a partial solution.
5.00
01 Jul 2007 - by Douglas Zare
Backgammon bot evaluations are useful, but rollouts more reliably estimate the true value of the position. A rollout may be untruncated (full) or truncated. What is the difference to us? Why would we prefer one type of rollout over another?
3.78
01 Jun 2007 - by Douglas Zare
What distinguishes backgammon from chess, go, bridge, and poker? Which skills are emphasized most in backgammon? Backgammon shares many features with other games, but it doesn't emphasize them in the same proportion. Instead of diving into positions and technical details this month, let's take a step back and consider how backgammon compares with other games in several categories.
4.78
07 May 2007 - by Douglas Zare
Deconvolution is a useful process in advanced mathematics. Convolution is blurring. Deconvolution is an attempt to recover the original sharp image or information. In this column, we'll look at natural deconvolution problems which occur in backgammon.
3.86
01 Apr 2007 - by Douglas Zare
In a holding game, your wins come from a combination of racing and hitting. How can we estimate the probability of hitting?
5.00
01 Mar 2007 - by Douglas Zare
Douglas Zare investigates the bias of backgammon bots, both of bots toward themselves, and of bots toward other bots. Along the way, he shows that bots disagree with each other more often and are farther from perfect play than you might expect.
4.80
01 Feb 2007 - by Douglas Zare
Before the dawn of backgammon bots, backgammon players would roll out positions by hand. Now, sophisticated backgammon bots do the rollouts automatically. But how do we know whether the bot is playing well in the rest of the game?
3.80
01 Jan 2007 - by Douglas Zare
Douglas Zare's latest article illustrates that objective numerical feedback is helpful, in backgammon and elsewhere in life. However, sometimes the feedback doesn't seem to make sense.
4.00
01 Dec 2006 - by Douglas Zare
Luck and skill are not opposites. That backgammon involves luck means that whether you have the advantage or not, your plays still matter. If you have fallen behind or blundered several times, your next play can affect the probability with which you win.
4.33
01 Nov 2006 - by Douglas Zare
There are several ways to quantify a skill advantage in backgammon, like error rates, ELO and expected points per game. There is no simple conversion between these measures, but in this column, we'll find some connections.
4.83
01 Oct 2006 - by Douglas Zare
In this column, we will introduce a numerical measure of the strength of a prime, blocking power. This is the number of pips you are forced to play elsewhere, not rolls. Like the effective pip count, or the probability of hitting a shot, blocking power must be combined with other considerations.
5.00
01 Sep 2006 - by Douglas Zare
Counting wins is a simple but powerful technique for approaching some tactical take/pass decisions, those likely to simplify quickly into positions that are easier to evaluate.
4.67
01 Aug 2006 - by Douglas Zare
Some serious observations about playing for money on a site with an Elo rating system can be applied by backgammon players below the expert level. You don't have to be a world-class player to win money playing backgammon.
4.00
01 Jul 2006 - by Douglas Zare
How do we measure the size of an error? How do we determine which errors are the most serious, and most worthy of attention? Two possibilities are to use match winning chances (mwc) and the money game equivalent (EMG). We'll see that neither is ideal, and that other normalizations are possible.
4.67
01 Jun 2006 - by Douglas Zare
When I make a really bad play, I study the position carefully until I understand the conceptual mistake I made. Afterwards, I sometimes get nightmares about my mistake.
4.00
01 May 2006 - by Douglas Zare
In Hoyle's 1745 work, "A Short Treatise On the Game of Back-Gammon," he covered many topics relevant to modern backgammon. In Chapter 7, he asked how likely it is to win a gammon after closing out one checker, when your opponent has 6 checkers to bring in plus the checker on the bar. His answer was that you win about 50% gammons.
4.71
01 Apr 2006 - by Douglas Zare
Archaeologists have told me about the old New Math. At some point in the last millennium, mathematicians tried to introduce layers of abstraction so that children could distinguish between the number 3 and a set containing 3.
4.89
01 Mar 2006 - by Douglas Zare
In this column, we'll look at the effective pip count of late bearoff positions - shorter than a 4-roll position. These are not hard to estimate with a little practice.
4.67
31 Jan 2006 - by Douglas Zare
In this column we'll look at potential doubles and redoubles in volatile positions where hitting is close to gin and missing usually leaves a race.
4.33
01 Jan 2006 - by Douglas Zare
Most features of rarely encountered Crawford and post-Crawford match scores can be understood by interpolating between a few common scores. We find that they are similar to each other and most decisions do not become increasingly extreme.
4.88
01 Dec 2005 - by Douglas Zare
'33% Speed' is a fast approximation to a normal 3-ply evaluation. Instead of averaging over all 441 dice sequences, the bot uses a balanced collection of 1/3 of the sequences and it is quite accurate for races and priming positions.
4.75
01 Nov 2005 - by Douglas Zare
In a mathematical approach to backgammon, we often want to convert one set of numbers to another and there are ways to construct accurate and simple mathematical models for these.
4.40
01 Oct 2005 - by Douglas Zare
Chouettes add fun and complication to the game of backgammon. If you simply make the best play at each oppportunity, you will do well, but there is room for improvement.
4.67
01 Sep 2005 - by Douglas Zare
At which match scores is the take point the most sensitive to the skill difference? We will study this question, and will find some answers that may be counterintuitive to most competitive players.
4.50
01 Aug 2005 - by Douglas Zare
It is not hard to count shots. The goal is to weigh the total cost of getting hit against the benefits when not hit. Let's consider the numerical cost of getting hit in a few situations and show numerical analysis done with the help of a bot.
4.20
01 Jul 2005 - by Douglas Zare
Simple racing formulas tell us the take point for efficient races, where neither side has buried checkers. In this column, we will find simple, important adjustments to the racing formulas to use in inefficient races.
4.13
01 Jun 2005 - by Douglas Zare
Is backgammon solvable by a database? If the entire game is too complicated, which aspects of the game can be solved by databases? To try to answer these questions, we need to count positions, and look at the storage capabilities of computers.
4.00
01 May 2005 - by Douglas Zare
Deep anchor games hit many shots in the bearoff. They also get gammoned a lot. The relative frequencies of these depend on many factors, but one of the most important is the race.
4.14
01 Apr 2005 - by Douglas Zare
The effective pip count can be useful for analyzing races to save the gammon. In holding games, running gammons start to be significant with a 60 pip lead, and a 77 pip lead may make the race for the gammon close.
4.43
01 Mar 2005 - by Douglas Zare
Good numerical feedback from bots makes it is easier to improve if you focus on numbers you can control in the short run rather than trying to affect the outcome of the game.
4.70
01 Feb 2005 - by Douglas Zare
While we consider the effective pip count for races, particularly in the bearoff, it can also be useful in contact positions that often result in races.
4.86
01 Jan 2005 - by Douglas Zare
Deep anchors generate many shots. In this column, we will take a quick survey of deep anchor games. We will look at how frequently they hit, how valuable those hits are, and the cost of keeping a deep anchor in backgammons.
3.67
01 Dec 2004 - by Douglas Zare
Let us consider some common positions with only two ways to win: win the race and hit a shot. The two are easy to analyze separately, and the pieces can be combined to give an estimate of the winning chances over the board.
4.25
01 Nov 2004 - by Douglas Zare
High-anchor holding games are common, so it is important to understand them. Let's consider one possible phase of a high-anchor holding game that can help us to make accurate plays game after game.
5.00
01 Oct 2004 - by Douglas Zare
Propositions arise when players disagree about the value of a position. Props are educational. Rollout results don't tell you how to play a position. You learn a lot by playing it out many times and it helps in decisions you'll face later.
4.80
01 Sep 2004 - by Douglas Zare
A freezeout is when two players bring money to a table and then play according to table stakes until one player wins all the money. Though more common in poker, here are some theoretical issues related to freezeouts in backgammon.
5.00
01 Aug 2004 - by Douglas Zare
The mathematical model of backgammon breaks down in interesting ways. In this column, we'll see what happens if you play out the Paradox Position many times. We'll also look at match play and table stakes.
5.00
02 Jul 2004 - by Douglas Zare
The doubling cube may escalate so rapidly that the theoretical average value of a position (equity) may not be defined. Backgammon is the only 2-player zero-sum game allowing this possibility.
4.67
01 Jun 2004 - by Douglas Zare
Some 1-point matches turn into a race quickly. Others are exciting protracted battles that produce decision after decision. Is there an objective way to distinguish straightforward matches from those with more challenging decisions?
4.56
25 Apr 2004 - by Douglas Zare
Streaks make a good story, so we often hear about them. To argue that the dice seem nonrandom, people report improbable streaks. In this column, we'll look at the theory behind streaks of random and nonrandom events.
4.29
25 Mar 2004 - by Douglas Zare
Backgammon is a race with obstacles. It is relatively easy to understand the race but hard to understand the obstacles, and how they affect the race. Though not the most common view of backgammon, it is particularly useful in holding games.
4.60
25 Feb 2004 - by Douglas Zare
My first trophy came from the Chouette Tournament at the 3rd Boston Open. I had faced some tough, unusual dilemmas. Here are a few of those along with some analysis of the tournament equity.
4.86
25 Jan 2004 - by Douglas Zare
In Part 1 we considered the theory of the bearoff near pure n-roll versus n-roll positions. In this second part, we consider actual examples of close cube decisions where one side is close to an n-roll position.
3.86
25 Dec 2003 - by Douglas Zare
Since high cubes are often exchanged at the end of the game, it is valuable to be able to judge the late bearoff accurately. Let's review the theory of 2-roll, 3-roll, and 4-roll positions and see the effects of each nonworking double or misses.
4.80
25 Nov 2003 - by Douglas Zare
The Kelly Criterion for rational gambling with an advantage is familiar to serious card counters in blackjack, but many strong backgammon players are not familiar with it.
4.78
25 Oct 2003 - by Douglas Zare
The mathematical theory of voting is surprisingly deep and some of the complexities show up in backgammon. This is not easy, and it is quite possible for a team to fail to make the most of its resources.
3.60
25 Sep 2003 - by Douglas Zare
Most of the time, having a feel for the position is more important than any calculation, but sometimes we need to count something. Here, we will focus on what to count, and when it is important to count.
4.95
25 Aug 2003 - by Douglas Zare
What distinguishes experts from each other and from advanced players, checker play or cube play?
4.70
25 Jul 2003 - by Douglas Zare
In money play, a perfectly efficient double is a borderline take/pass. Cube vigorish can be used to measure the efficiency of doubles.
5.00
25 Jun 2003 - by Douglas Zare
Closeouts can lead to the gammons but are also a route to victory when you hit your opponent after he has started to bear off. Let's look at some of the resulting positions, and two heuristics, to help you to estimate winning chances.
4.82
26 May 2003 - by Douglas Zare
There are many formulas that can be used to try to reconstruct the match equity table over the board. These include Janowski's formula and Neil's numbers. Here's a new method, the 2:1 rule, that covers some additional cases.
4.67
25 Apr 2003 - by Douglas Zare
The pip count is a simple way to assess most races, including races to save the gammon but in many situations, the raw pip count is unsatisfactory, and must be adjusted.
4.81
25 Mar 2003 - by Douglas Zare
I have often looked over positions from matches of mine and been dismayed by a bad take/pass, but thought, "At least it would be a take/pass for money." Well, how much of a difference does it make? Quite a lot!
4.17
25 Feb 2003 - by Douglas Zare
Woolsey's Rule is supposed to encourage you to double when doubling is correct, but here we'll concentrate on doubling when it is wrong, bluffing.
3.89
25 Jan 2003 - by Douglas Zare
Just as musicians practice making a single note beautiful, backgammon players should benefit from asking, "What is the purpose of doubling?"
4.56
25 Dec 2002 - by Douglas Zare
If someone offers to flip a coin to decide whether to take or not, double without hesitation.
5.00
25 Nov 2002 - by Douglas Zare
The basic idea of parity is easy, but many serious players err in the execution, applying parity too much or at the wrong time.
5.00
22 Oct 2002 - by Douglas Zare
Rollouts are one of the most powerful tools for assessing the value of a position. This article intends to help illustrate what information we can or can't get from them, and help guide the process of choosing which rollouts to perform.
4.89
22 Sep 2002 - by Douglas Zare
There are some very simple calculations based on the normal distribution that help one to understand results from rollouts and money sessions.
4.56
22 Aug 2002 - by Douglas Zare
In this article, we will consider Janowski's formulas for cubeful equity when not too good to redouble and the redoubling point.
4.20
22 Jul 2002 - by Douglas Zare
Volatility is about how much is riding on the next roll or exchange and when to double - it is usually correct that you need an advantage and a volatile position, but sometimes this is misleading.
4.88
22 Jun 2002 - by Douglas Zare
In Snowie's theory panel, you can find a lot of information about match play according to Snowie's match equity table. One statistic for a match score and cube level is the early-late ratio. What is this ratio and how can you use it?
4.92
22 May 2002 - by Douglas Zare
Rick Janowski has created many formulas relevant to backgammon. As we look at the Janowski take point formula for money play, we will see the continuous limit model of backgammon and cube liveliness.
4.60
22 Apr 2002 - by Douglas Zare
In pure mathematics, the reflection principle is a clever idea that sometimes produces simple solutions to hard problems, or relates apparently different phenomena.
4.27
22 Mar 2002 - by Douglas Zare
As I try to understand aggression in backgammon, let's focus on checker plays in positions resulting from closeouts that many players get wrong. Although these errors are not individually very costly they do arise frequently.
4.71
22 Feb 2002 - by Douglas Zare
Tournaments thrill the participants and spectators. But is there any theory involved? Surely the correct strategy is just to play your best in each match, right? Let's look at this a little deeper.
4.43
22 Jan 2002 - by Douglas Zare
Bots have positive and negative confusion levels.
4.36
22 Dec 2001 - by Douglas Zare
Is match play like money play if both players can use all of the points?
4.11
22 Nov 2001 - by Douglas Zare
Gammon price is how valuable it is to win at the current cube level but how valuable it is relative to the value of winning a single game rather than losing.
4.83
