Somebody offers you a deal that is worth taking and feels insulting. You turn it down, and you are worse off for having done so. This article is about the most studied experiment in behavioural economics, which is about exactly that moment, and about why the behaviour is more strategic than it looks.
Key Takeaway
A meta-analysis of 37 papers with 75 results finds that "on average the proposer offers 40% of the pie to the responder" and that "on average 16% of the offers is rejected."[1] Our own arithmetic: rejecting a 5 percent offer destroys nineteen dollars of the proposer's money for every dollar of your own, so the worse the offer, the cheaper the punishment.
The Verdict, Stated First
Five claims, in descending order of confidence.
One. The core result is as solid as anything in this series. A meta-analysis across 37 papers and 75 results, on an experiment simple enough that nothing can go wrong in the design.
Two. The punishment mechanism is self-sharpening and this is arithmetic, not psychology. Leverage rises from 1 to 1 at an even split to 19 to 1 at a five percent offer, so rejection gets cheaper exactly where offers get worse.
Three. Proposers are more generous than pure self-interest requires, and not by much. On our own illustrative curves a cold calculator offers 23 to 32 percent against an observed 40, a gap of eight to seventeen points.
Four. The generosity is nearly free, which is the practically useful finding. On our own arithmetic, moving from a 28 percent offer to 40 costs the proposer about four dollars on a hundred dollar pie while raising acceptance from 81 to 91 percent.
Five. Cross-cultural differences are real and do not map onto cultural theories. The meta-analysis finds differences in responder behaviour across regions which, with one exception, "cannot be attributed to various cultural traits" underlying the standard classifications.
Our Grades For These Claims
Applying the scheme from the first article in this series.
Grade A for the meta-analytic figures, from an abstract obtained verbatim from six independent sources including the publisher.
Grade A for the game's structure and standard result, obtained verbatim from an academic preprint quoting the literature.
Grade B for the cross-cultural extremes, which come from the meta-analysis's own working paper describing two studies we did not obtain, both with very small samples.
Grade A for our own arithmetic, which is elementary, though every parameter in the optimal-offer section is invented.
Grade C for anything about motivation, since the game's own inventor wrote thirty-two years later that the motivations are diverse.
Our position: the behaviour is extremely well established and the explanation for it is not, which by now is the least surprising sentence in this series.
A Note On Method
Everything here is verified to August 2026.
We obtained the 2004 meta-analysis's abstract verbatim from six independent sources, including the publisher[1], and passages of its working paper version[2]. We did not obtain the published paper.
We obtained the game's structure and the standard empirical result verbatim from an academic preprint quoting the literature[3]. We did not obtain the 1982 paper.
The cross-cultural figures come from the meta-analysis's own working paper describing two other studies[2], and we obtained neither of those studies.
We did not obtain any of the Henrich papers, the 2014 review by the game's inventor, or any of the theoretical literature, and report all from citation records and one quoted phrase.
All arithmetic is ours. The leverage calculation is trivial; the optimal-offer section uses an invented rejection curve throughout and our first attempt at it produced a range we had to correct, which we describe in the body.
This article discusses research on bargaining. It is not negotiation, legal or commercial advice.
The Game
The setup, in the clearest statement we found.
"There are two players and a pie. The first player, named 'proposer', has to decide how to divide the pie between herself and the second player, named 'responder'. At the same time, the responder decides his minimum acceptable offer (MAO). If the proposer's offer is higher than or equal to the responder's MAO, then the pie is divided between the proposer and the responder as agreed; otherwise, no one get anything."[3]
The source is Güth, W., Schmittberger, R., and Schwarze, B. (1982), An experimental analysis of ultimatum bargaining, Journal of Economic Behavior and Organization, 3(4), 367–388[4].
Three observations, ours.
The design is one-shot and anonymous in the standard version, which removes reputation entirely. Nothing you do can affect a future round, because there is not one.
The pie is destroyed on rejection, not returned. That is what makes refusal costly rather than merely stubborn, and it is the feature every business analogy has to check before borrowing the result.
And the scale of the literature is worth one line. A 2018 preprint records that searching for the game "generates over 58,000 results" and that the 1982 paper had "collected over 4,000 citations"[3].
What Theory Predicts
The benchmark the result is measured against.
"A self-interested responder would accept any non-zero offer; anticipating this behavior, a self-interested proposer would offer the smallest positive amount possible."[3]
Three observations, ours.
The prediction is not a caricature. It follows in one step from two assumptions economics made explicitly, and the argument is airtight given them.
It also has a feature worth noticing. The responder's decision comes first logically and second in time, and everything the proposer does depends on predicting it.
So the whole result is really about the responder. If responders accepted everything, proposers would offer nothing, and the interesting behaviour is on the receiving end.
What Actually Happens
The standard finding, stated by a preprint summarising six sources.
"These predictions are blatantly violated in laboratory experiments: offers below 25% are typically rejected, and the vast majority of offers lie between 30% and 50%."[3]
Four observations, ours.
Both halves of the prediction fail. Proposers do not offer the minimum and responders do not accept it, and the second failure explains the first.
The threshold is specific. Below 25 percent, offers are typically rejected, which gives a business reader a number rather than a sentiment.
And the range 30 to 50 percent is narrow. This is not a finding about people varying wildly; it is a finding about most people converging on something close to a third or a half.
Note that the summary cites six sources spanning 1982 to 2014, which is the kind of corroboration this series usually cannot report.
The Meta-Analysis
The quantitative synthesis.
Oosterbeek, H., Sloof, R., and van de Kuilen, G. (2004), Cultural Differences in Ultimatum Game Experiments: Evidence from a Meta-Analysis, Experimental Economics, 7(2), 171–188, June, DOI 10.1023/B:EXEC.0000026978.14316.74[1].
Its abstract: "This paper reports the findings of a meta-analysis of 37 papers with 75 results from ultimatum game experiments. We find that on average the proposer offers 40% of the pie to the responder. This share is smaller for larger pie sizes and larger when a strategy method is used or when subjects are inexperienced. On average 16% of the offers is rejected. The rejection rate is lower for larger pie sizes and for larger shares offered. Responders are less willing to accept an offer when the strategy method is employed."[1]
We obtained the abstract from six independent sources and not the paper.
Forty Percent And Sixteen Percent
The two numbers to carry. Ours.
Four observations.
Forty percent is the average offer. Against a theoretical prediction of approximately zero, that is not a modest deviation; it is a different answer.
Sixteen percent is the average rejection rate. That means roughly one bargain in six fails, and the pie is destroyed each time.
The two figures are in tension in a useful way. If proposers are offering 40 percent, and 16 percent of offers still get rejected, then a meaningful share of responders are turning down close to a third of a pie.
And both numbers are averages across widely varying conditions, which the abstract's next sentences immediately qualify.
The Moderators
What moves the two figures, and one of them is directly commercial.
Three observations, ours.
Offers are smaller and rejections rarer for larger pie sizes. Both effects point the same way: as the money gets serious, behaviour moves toward the self-interested prediction.
That is the single most commercially relevant moderator in the abstract, because business bargains are large relative to laboratory pies. The experimental literature may systematically overstate how much fairness survives at commercial stakes.
And the strategy method moderator is methodological rather than behavioural. Asking a responder in advance which offers they would accept produces larger offers and more rejection than letting them respond to a real one, which is a warning about how the question is put.
Responders, Not Proposers
The cross-cultural finding, stated precisely.
"We find differences in behavior of responders (and not of proposers) across geographical regions."[1]
Four observations, ours.
The asymmetry is the finding and it is easy to read past. Proposers behaved similarly everywhere. Responders did not.
That fits the structure of the game. Proposers are solving a prediction problem and responders are expressing a preference, and preferences are what vary between places.
It also implies something uncomfortable for proposers. If offer behaviour is universal and acceptance behaviour is not, then proposers in some places are systematically mispredicting their counterparts.
And we obtained no magnitudes, so we cannot tell you how large the regional differences were.
And Not Explained By Cultural Classifications
The negative result inside the same abstract, which is the part we would put first.
"With one exception, these differences cannot be attributed to various cultural traits on which for instance the cultural classifications of Hofstede (1991) and Inglehart (2000) are based."[1]
The working paper version puts it differently: the differences "tend to follow geographical lines rather than the cultural classifications provided by Hofstede (1991), and by Huntington (1996) and Inglehart (2000) respectively."[5]
Four observations, ours.
This is a failure of the standard cultural frameworks to predict the one thing that varies, reported by authors who set out to test them.
The distinction between geographical lines and cultural traits is doing real work. Neighbouring places behave similarly whether or not the classification schemes group them together.
Which is a warning for anyone using those frameworks commercially. A dimension score did not predict bargaining behaviour, in the one study we found that checked.
And note the difference between versions. The working paper names three classification systems and the published abstract names two, dropping one. We report both and treat neither as an error.
Twenty-Six And Fifty-One
The range across societies, from the meta-analysis's own working paper.
It records: "Henrich (2000), for instance, reports an average offer equal to 26% among 21 participants in Peru who each had to divide USD 160 with their opponent. In contrast, Buchan et al (1999) find among 11 participants in Japan an average offer of 51% when the total pie equals USD 50."[2]
It adds: "Henrich et al (2001) find substantial differences in the outcomes of the ultimatum game between subjects in 15 different societies."[2]
We obtained none of these studies and report the descriptions.
Three observations, ours.
Twenty-six percent against fifty-one is nearly a factor of two, and the low end sits at the threshold below which offers are typically rejected in the standard literature.
The Peruvian pie was USD 160 against the Japanese USD 50, and the abstract above says offers are smaller for larger pies. Some of that gap is pie size rather than culture, and we cannot tell you how much.
And that confound is exactly why a meta-analysis exists. The whole point of pooling is to separate the moderators, which two individual studies compared side by side cannot do.
Twenty-One People And Eleven People
The observation we think matters most about those figures. Ours.
Four observations.
Twenty-one participants and eleven participants. Those are the samples behind the two numbers most often cited to show that fairness norms vary between societies.
An average of eleven observations carries an enormous confidence interval, and this series has computed the general form of that before. The sixty-sixth article showed a rate from four observations carrying an interval of roughly fifty points.
We are not saying the cross-cultural finding is wrong. The meta-analysis pools 75 results and finds regional differences in responders, which is much stronger evidence than any pair of studies.
We are saying the headline anecdote is thinner than the finding it illustrates, which is a pattern this series has now recorded in a dozen literatures. The vivid example and the actual evidence are rarely the same thing.
The Null Result Nobody Quotes
A study on the other side, from the same working paper.
"In contrast, Okada and Riedl (1999) find no differences in offers or rejection rates between subject groups in Vienna and Kyoto."[2]
We did not obtain it and report the sentence.
Three observations, ours.
Vienna and Kyoto are about as culturally distant as two developed cities get, and one study found no difference in either offers or rejections.
That is entirely consistent with the meta-analysis. Differences follow geographical lines and not cultural traits, so two distant-but-comparable urban populations behaving alike is what that finding predicts.
And it is the kind of result that does not travel. The Peruvian and Japanese figures are quoted constantly and this one is not, which is the ordinary asymmetry between findings and non-findings.
Punishment Leverage
The arithmetic feature that explains the whole result. Our own calculation, on a hundred dollar pie.
Rejecting an offer of X means giving up X to deny the proposer the rest. The leverage is the ratio.
At a 50 percent offer: you forgo $50 to deny $50. Leverage 1.0 to 1.
At 40: $40 against $60. Leverage 1.5 to 1.
At 30: $30 against $70. Leverage 2.3 to 1.
At 25: $25 against $75. Leverage 3.0 to 1.
At 20: $20 against $80. Leverage 4.0 to 1.
At 10: $10 against $90. Leverage 9.0 to 1.
At 5: $5 against $95. Leverage 19.0 to 1.
Why The Mechanism Sharpens Itself
Reading that table. Ours, and we have not seen the point made in these terms.
Four observations.
The worse the offer, the cheaper it is to punish. Leverage rises from parity at an even split to nineteen to one at five percent, and it is a smooth function of how unfair the offer is.
So refusal is most affordable exactly where it is most warranted, which means the threat needs no irrationality to be credible. A responder willing to spend five dollars can impose ninety-five.
This also explains the empirical threshold. Offers are typically rejected below 25 percent, which on this table is where leverage passes three to one, and above which punishment starts getting expensive relative to what it destroys.
And it explains why the rejection rate falls with pie size, as the abstract reports. Leverage is a ratio and the sacrifice is an absolute amount, so on a large enough pie even a nineteen-to-one punishment costs real money.
That last point deserves working through, because it is the one a business reader should carry into a real deal. The leverage ratio is scale-free and the sacrifice is not. Turning down five percent of a hundred dollars costs five dollars; turning down five percent of two million costs a hundred thousand.
So the same insult produces different behaviour at different scales, and the direction is predictable. As stakes rise, the punishment gets more satisfying in ratio terms and more expensive in absolute terms, and the meta-analysis reports the absolute term winning.
Which gives a rough working rule we would state cautiously. Expect the fairness constraint to bind hardest on small, frequent transactions and to weaken on large, rare ones, which is close to the opposite of how most firms allocate their attention to it.
What A Cold Proposer Should Offer
The question a business reader actually has. Our own arithmetic, with an entirely invented rejection curve of the form: probability of rejection falls exponentially with the size of the offer.
Under a steep curve, a purely self-interested proposer offers 23.0 percent and expects to keep $67.28.
Under a moderate curve, 27.9 percent, keeping $58.58.
Under a shallow curve, 31.3 percent, keeping $51.90.
Three observations.
Even a proposer who cares nothing for fairness offers a substantial share, because the punishment threat is real and rejection destroys everything.
So a large part of the observed 40 percent is not generosity at all. It is a correct response to a counterparty who will burn the deal.
And the range across our three curves is 23 to 32 percent, which is where our own draft went wrong.
We Got The Range Wrong
Continuing a practice from the last nine articles, because this one failed in the most ordinary way.
Our first draft of the section above stated that a self-interested proposer would offer "between 26 and 45 percent." The computation gives 23.0, 27.9 and 31.3.
Three observations, ours.
The error is that we wrote the sentence before running the curve. We had an expectation that the optimum would bracket the observed 40 percent, wrote that, and the arithmetic did not support it.
Catching it changed the conclusion rather than tidying it. The corrected range sits entirely below the observed average, which means proposers are more generous than self-interest requires, not merely close to it.
And the corrected version is more favourable to the behavioural reading than our draft was, which is worth stating because a reader is entitled to wonder whether we correct only in the convenient direction.
The Gap That Remains
What the corrected figures actually show. Our own arithmetic, invented parameters.
Against an observed average offer of 40 percent: the steep curve's optimum is 17.0 points below, the moderate curve's 12.1 points below, and the shallow curve's 8.7 points below.
The cost of that extra generosity to the proposer: $8.92, $4.02 and $1.82 respectively, on a hundred dollar pie.
Four observations.
The gap is real on all three curves. Strategic fear alone does not explain the observed 40 percent, and something else is operating.
But the gap is modest and it costs very little, which is the finding we did not expect and which reframes the whole debate.
Most writing on this experiment treats it as a contest between selfishness and fairness. The arithmetic says the two answers are within a few dollars of each other, which makes the contest less consequential than it reads.
And every parameter here is ours. A different rejection curve moves every number, and no source we obtained supplies one.
One further caution about the shape of our curve, since we chose it. An exponential rejection function assumes rejection falls smoothly and never quite reaches zero, which is a modelling convenience rather than a fact. The literature reports a threshold at around 25 percent, and a curve with a genuine cliff at that point would produce a different and probably higher optimum.
The Flat Maximum
The most useful thing in this article, and it fell out of correcting our own error. Our own arithmetic, moderate curve, invented parameters.
Expected value to the proposer, by offer: at 20 percent, acceptance 69.9 percent, keeping $55.90. At 25: 77.7 percent, $58.27. At 28: 81.4 percent, $58.58. At 32: 85.3 percent, $58.03. At 36: 88.5 percent, $56.62. At 40: 90.9 percent, $54.56. At 50: 95.0 percent, $47.51.
Four observations.
The curve is almost flat between 25 and 32 percent, varying by about thirty cents across that range. There is no sharp optimum to find.
Moving from the optimum at 28 to the observed 40 costs $4.02 and buys 9.5 percentage points of acceptance. That is cheap insurance against a failed deal.
Which is the practical lesson and it is not about fairness. Near the optimum, extra generosity is nearly free and buys a materially higher chance of closing, so a proposer optimising hard is optimising a flat surface.
And that connects to the sixty-seventh article's argument. Where the payoff surface is flat, precision in the rule matters little and consistency matters more, which is the same reason simple models do well.
Motivations Are Diverse
The inventor's own verdict, thirty-two years later.
Reference lists identify Güth, W., and Kocher, M. (2014), How Werner Güth's ultimatum game shaped our understanding of social behavior, Journal of Economic Behavior and Organization, 108, 292–318[6], described by a citing preprint as "their exceptional review of thirty years of UG experiments" and quoted as concluding that "motivations behind decisions in the ultimatum game are diverse."[3]
We did not obtain the review and report the quoted phrase.
Four observations, ours.
The first-named author is the same Güth who ran the original experiment, writing thirty-two years on, and his summary is that the motivations are diverse rather than identified.
Candidates in the literature we can see from reference lists include envy, social comparison, property rights and anonymity, and social value orientation[6][7]. We obtained none of those papers.
This is now the seventh literature in which this series has found a well-measured behaviour with a contested mechanism, and we would treat that as the default expectation rather than a surprise.
And the practical consequence is the usual one. The arithmetic of leverage works whatever the motivation is, so a reader can use this article without the dispute being settled.
Five Variants Of Two Citations
The bibliographic entry, since this series keeps a running count and this topic produced the densest cluster yet.
The 1982 paper's third author, Bernd Schwarze, appears in one journal's reference list as "Schwartz, R.", which is a different surname and a different initial[7]. Another source renders the first author as "Güth W.R.", adding a middle initial[8].
The 2001 fifteen-societies paper appears with pages 73–78 in most sources and 73–79 in the 2014 review[6]; with its author order given as Camerer, Fehr, Gintis, McElreath in most and Camerer, Gintis, McElreath, Fehr in that review; and with its title carrying the word "Behavioral" in most and omitting it in that review[6].
Three observations, ours.
None of the five changes anything findable. Volumes, years and journals are correct throughout.
The author-order variant is the most consequential, because author order carries credit in economics and a reader citing from that list would misattribute it.
That brings the running count of bibliographic variants across this series to twenty-two, and we note the 2014 review is by the experiment's own inventor.
Where The Fairness Actually Sits
A restatement of the whole finding that follows from the arithmetic and that we have not seen made. Ours.
The experiment is universally described as showing that people are fairer than economics assumes. The arithmetic locates that fairness in one specific place.
Four observations.
The proposer's behaviour is mostly explained without fairness at all. On our own curves a purely self-interested proposer offers 23 to 32 percent, which is a substantial share arrived at by pure calculation.
The responder's behaviour is not. Refusing a real payment to punish an anonymous stranger you will never meet again cannot be rationalised by self-interest under any curve, because the game ends.
So the honest summary is narrower and stronger than the usual one. The proposer is being prudent and the responder is being fair, and it is the second that constitutes the finding.
And that matches the cross-cultural result exactly, which is the part that persuaded us. The meta-analysis found differences in responders and not in proposers, which is precisely what you would expect if proposer behaviour is a calculation and responder behaviour is a preference. Two independent lines pointing at the same conclusion.
One Machinery, Two Articles
How this sits against the sixty-fourth article, because the two findings are usually run together and should not be. Ours.
Four observations.
That article reported what people say about pricing, from telephone surveys of hypothetical scenarios, and reported one study finding those judgments did not predict behaviour.
This one reports what people do with real money at stake, and the doing is costly and measured.
The two together are more useful than either. The survey work establishes the content of the norm, being that cost increases may be passed on and demand increases may not. The bargaining work establishes that people will pay to enforce a norm.
What neither establishes is the link. We found no study showing that customers who judge a price unfair then impose costly punishment, and that is the study a business owner would most want. We flag its absence rather than assuming the two literatures join up.
What Actually Survives
Our reading, stated directly.
Five statements.
The behaviour is as well established as anything in this series. Meta-analysis across 37 papers and 75 results: 40 percent offered, 16 percent rejected.
The punishment mechanism is arithmetic and self-sharpening, with leverage rising from parity at an even split to nineteen to one at five percent.
Most of the observed generosity is strategic, on our own curves, with a residual gap of eight to seventeen points that costs the proposer between two and nine dollars on a hundred.
Cross-cultural differences appear in responders and not proposers, and do not map onto the standard cultural classifications.
And the motivation remains unidentified, on the verdict of the person who invented the experiment.
Your Negotiations
The first application. Ours, untested, and not negotiation or commercial advice.
Four points.
The leverage table is the thing to internalise. A counterparty you have squeezed to five percent of the surplus can destroy nineteen dollars of yours for every dollar they give up, and they do not need to be irrational to do it.
So the risk of an aggressive offer is not that it will be refused and renegotiated. It is that refusal is cheap for the other side at exactly the point where you have taken the most.
The empirical threshold gives a working number. Below roughly 25 percent of the visible surplus, expect refusal to become likely, in the laboratory at least.
And the moderator cuts the other way at scale. Rejection rates fall as pies get larger, so the laboratory threshold probably overstates the risk on a substantial commercial deal.
The Last Twelve Points
The second application, and the one we would actually use. Ours.
Four points.
On our own moderate curve, moving an offer from 28 percent to 40 costs the proposer $4.02 on a hundred dollar pie and raises acceptance from 81.4 to 90.9 percent.
In deal terms that is four percent of the pie for nine and a half points of closing probability, which is a trade most people would take if it were presented as a number.
It is almost never presented that way. Negotiators optimise the split and treat closing probability as a separate worry, when the two are the same calculation.
And the general form is the flat maximum. Near the optimum you are giving up very little for each point conceded, so the last few points of a hard-fought split are usually the cheapest thing you own.
Two cautions on using that, because it is the most actionable claim in this article and it rests entirely on parameters we invented. The flatness is a property of our curve rather than a measured fact, and a rejection function with a sharp threshold would produce a peak rather than a plateau.
And the argument runs in the other direction too, which is worth saying to anyone tempted to concede reflexively. If the surface really is flat, then conceding the last twelve points buys very little as well as costing very little, and the case for it rests on the acceptance gain rather than on the money being trivial.
What Sixteen Percent Costs
The third application, and the one that scales. Our own arithmetic, invented figures.
At the meta-analytic rejection rate of 16 percent, on a pie worth $50,000:
10 negotiations produce 1.6 failures and destroy $80,000 of joint surplus. 50 produce 8.0 and destroy $400,000. 100 produce 16.0 and destroy $800,000. 250 produce 40.0 and destroy $2,000,000.
Three observations.
The destroyed value is joint, not one side's. Both parties lose it, which is what makes the mechanism expensive and what makes it work.
The laboratory rate is almost certainly too high for commercial deals, since rejection falls with pie size and real deals allow counteroffers. We use it because it is the figure that exists.
And the point is the order of magnitude rather than the number. A failure rate in the mid-teens across a book of negotiations destroys more value than most firms would tolerate anywhere else, and it is rarely measured at all.
One Difference From Your Situation
The caveat that limits everything above, and it is substantial. Ours.
Four observations.
The standard game is one-shot, anonymous, and take-it-or-leave-it. Almost no business negotiation is any of those three.
Repetition changes the analysis entirely. A responder in an ongoing relationship has reputational reasons to refuse that the experiment removes, which should make refusal more likely rather than less.
But the counteroffer changes it the other way, and more strongly. A real negotiation rarely destroys the pie on a single refusal; it produces a revised offer, which is precisely what the experiment forbids.
So the honest transfer is narrow. The leverage arithmetic holds wherever refusal genuinely destroys the surplus, being a walked-away deal, a lost client, a resignation. Where a counteroffer is available, the experiment describes the worst case rather than the expected one.
What To Do
Learn the leverage table. At a five percent share a counterparty destroys nineteen dollars of yours per dollar of their own; at an even split, one for one. Punishment is cheapest exactly where your offer is worst.
Treat 25 percent of visible surplus as a threshold. Below it, offers are typically rejected in the laboratory literature, and that is a number rather than a sentiment.
Price the last few points of a split. On our own curve, conceding from 28 to 40 percent costs four percent of the pie and buys 9.5 points of closing probability.
Do not optimise a flat surface. Expected value varies by about thirty cents across offers from 25 to 32 percent on our own figures, so hard bargaining in that range is effort spent on nothing.
Expect fairness to weaken as stakes rise. The meta-analysis reports smaller offers and fewer rejections for larger pies, both moving toward the self-interested prediction.
Do not use cultural frameworks to predict bargaining. Regional differences appeared in responders and, with one exception, could not be attributed to the traits underlying the standard classifications.
Check whether refusal really destroys the pie. Where a counteroffer is available the experiment describes a worst case; where it is not, the leverage arithmetic applies directly.
Measure your own rejection rate. At the laboratory figure of 16 percent, a hundred negotiations on a fifty thousand dollar pie destroy eight hundred thousand dollars of joint surplus, and almost nobody counts it.
The Limits Of This Analysis
Several caveats matter. This article discusses research on bargaining and is not negotiation, legal or commercial advice; the applications are our own reasoning and untested. Everything is verified to August 2026. We did not obtain the 2004 meta-analysis, only its abstract from six independent sources plus passages of its working paper version, so we report no magnitudes for the regional differences it found and no confidence intervals on the 40 and 16 percent figures. We did not obtain the 1982 paper and report the game's structure and the standard result through an academic preprint quoting six sources. We obtained none of the cross-cultural studies, and the 26 and 51 percent figures reach us through the meta-analysis's working paper describing them; those two studies had 21 and 11 participants respectively and their pies differed in size, which the abstract's own moderator says affects offers, so part of that gap is not culture and we cannot say how much. We did not obtain the 2014 review by the experiment's inventor and report one quoted phrase from it, nor any of the papers on competing motivations. All arithmetic is ours. The leverage calculation is trivial and exact. The optimal-offer, gap and flat-maximum sections use an entirely invented exponential rejection curve with three invented constants; no source we obtained supplies a rejection function, and a different curve moves every figure in those three sections. Our first draft of the optimal-offer range was wrong, stating 26 to 45 percent where the computation gives 23 to 32, and we describe the error and its correction in the body. The surplus-destruction figures use an invented pie size and deal count. And the standard experiment is one-shot, anonymous and forbids counteroffers, which is true of almost no business negotiation; where a counteroffer is available, this literature describes a worst case rather than an expected one.
Frequently Asked Questions
What is the ultimatum game?
Why do people reject money?
Are proposers being generous?
What is the most useful practical finding?
Does culture explain the variation?
How well do the famous cross-cultural numbers hold up?
Does this transfer to real negotiations?
References
- Oosterbeek, H., Sloof, R., & van de Kuilen, G. (2004). Cultural Differences in Ultimatum Game Experiments: Evidence from a Meta-Analysis. Experimental Economics, 7(2), 171–188, June, DOI 10.1023/B:EXEC.0000026978.14316.74. Publisher record reproducing the abstract in full: on the paper reporting the findings of a meta-analysis of 37 papers with 75 results from ultimatum game experiments; on the authors finding that on average the proposer offers 40 percent of the pie to the responder; on that share being smaller for larger pie sizes and larger when a strategy method is used or when subjects are inexperienced; on 16 percent of offers being rejected on average; on the rejection rate being lower for larger pie sizes and for larger shares offered; on responders being less willing to accept an offer when the strategy method is employed; on meta-analysis providing an alternative way to investigate whether bargaining behavior differs across countries; on the authors finding differences in behavior of responders and not of proposers across geographical regions; and on those differences, with one exception, not being attributable to various cultural traits on which the cultural classifications of Hofstede (1991) and Inglehart (2000) are based. The abstract is reproduced identically by five further independent sources including two bibliographic services, a university repository, a publisher aggregator and a second publisher platform. Note: the publisher's record, corroborated by five independent reproductions. We obtained the abstract only and not the paper, so we report no magnitudes for the regional differences and no confidence intervals. link.springer.com
- Working paper version of the same meta-analysis, hosted on an economics working paper archive, reproducing passages of its introduction: recording that a key insight from over two decades of experimental economics research is that people typically do not behave as selfishly as traditional economics assumes; that Henrich (2000) reports an average offer equal to 26 percent among 21 participants in Peru who each had to divide USD 160 with their opponent; that in contrast Buchan and colleagues (1999) find among 11 participants in Japan an average offer of 51 percent when the total pie equals USD 50; that Henrich (2000) reports differences between 18 to 30 year old Machiguenga men of the Peruvian Amazon and students at UCLA, with the former offering smaller amounts; that Henrich and colleagues (2001) find substantial differences in ultimatum game outcomes between subjects in 15 different societies; that in contrast Okada and Riedl (1999) find no differences in offers or rejection rates between subject groups in Vienna and Kyoto; and that the approach adopted in the cross-country and cross-culture studies has two potential drawbacks, at which point our reproduction ends. Note: a working paper version on an economics archive. Our source for the cross-cultural figures and the null result; we obtained none of the studies it describes, and the two headline figures rest on samples of 21 and 11 participants with differently sized pies. econwpa.ub.uni-muenchen.de
- Academic preprint on social versus moral preferences in the ultimatum game, reproducing the game's structure and the standard empirical result: that thirty-five years after its invention a search for the game generates over 58,000 results and the 1982 paper has collected over 4,000 citations; that there are two players and a pie, with the proposer deciding how to divide it and the responder deciding his minimum acceptable offer, the pie being divided as agreed if the offer meets or exceeds that minimum and otherwise nobody getting anything; that a self-interested responder would accept any non-zero offer and, anticipating this, a self-interested proposer would offer the smallest positive amount possible; that these predictions are blatantly violated in laboratory experiments, with offers below 25 percent typically rejected and the vast majority of offers lying between 30 and 50 percent, citing Camerer (2003), Fehr and Schmidt (1999), Fehr and Fischbacher (2003), Güth, Schmittberger and Schwarze (1982), Güth and Kocher (2014) and Henrich and colleagues (2001); and that Güth and Kocher, in their review of thirty years of experiments, conclude that motivations behind decisions in the ultimatum game are diverse. Note: an academic preprint quoting the literature. Our source for the game's structure, the standard result and the inventor's own verdict on motivation; we obtained none of the six works it cites for the empirical claim. arxiv.org
- Reference list carried on a peer-reviewed journal article, confirming Güth, W., Schmittberger, R., and Schwarze, B. (1982), An experimental analysis of ultimatum bargaining, Journal of Economic Behavior and Organization, 3(4), 367–388; Roth, A. E., Prasnikar, V., Okuno-Fujiwara, M., and Zamir, S. (1991), Bargaining and market behavior in Jerusalem, Ljubljana, Pittsburgh, and Tokyo, The American Economic Review, 81, 1068–1095; Henrich, J., Boyd, R., Bowles, S., Camerer, C., Fehr, E., and colleagues (2001), The American Economic Review, 91, 73–78; and Henrich, J., Ensminger, J., McElreath, R., Barr, A., Barrett, C., and colleagues (2010), Markets, religion, community size, and the evolution of fairness and punishment, Science, 327, 1480–1484. Note: a reference list carried on a peer-reviewed article, used to confirm the 1982 citation independently. We obtained none of the works named. pubmed.ncbi.nlm.nih.gov
- Working paper record for the same meta-analysis on a preprint service, reproducing an earlier version of its abstract which reports a meta-analysis of 32 papers rather than the published 37 with 75 results, and which states that the differences found tend to follow geographical lines rather than the cultural classifications provided by Hofstede (1991), and by Huntington (1996) and Inglehart (2000) respectively. Note: a preprint service record of an earlier version. Reported to show how the paper's own account changed between working and published versions, in the number of papers pooled and in the classifications named; we treat neither version as an error. papers.ssrn.com
- Repository copy of Güth, W., and Kocher, M. (2014), How Werner Güth's ultimatum game shaped our understanding of social behavior, Journal of Economic Behavior and Organization, 108, 292–318, carrying its reference list, which gives Henrich, J. (2000), Ultimatum game bargaining among the Machiguenga of the Peruvian Amazon, American Economic Review, 90, 973–979; Henrich, J., Boyd, R., Bowles, S., Camerer, C., Gintis, H., McElreath, R., and Fehr, E. (2001), In search of homo economicus: experiments in 15 small-scale societies, American Economic Review, 91(2), 73–79; Henrich and colleagues (2005), 'Economic man' in cross-cultural perspective: ethnography and experiments from 15 small-scale societies, Behavioral and Brain Sciences, 28, 795–855; and Henrich and colleagues (2004), Foundations of Human Sociality, Oxford University Press. Note: a repository copy of a review by the experiment's own inventor. Recorded as the source of three bibliographic variants in one citation: page range 73 to 79 against 73 to 78 elsewhere, a different author order, and a title omitting the word "Behavioral". We obtained the reference list and not the review's body. academia.edu
- Journal article page for a meta-study of responder behaviour in ultimatum games, carrying a reference list which renders the 1982 paper's authors as Guth, W., Schmittberger, R., and Schwartz, R., giving the third author a different surname and initial from the Schwarze, B. recorded everywhere else; and confirming Hoffman, E., McCabe, K., Shachat, K., and Smith, V. (1994), Preferences, property rights and anonymity in bargaining games, Games and Economic Behavior, 7(3), 346–380; and Kirchsteiger, G. (1994), The role of envy in ultimatum games, Journal of Economic Behavior and Organization, 25(3), 373–389. Note: a journal article's reference list. Recorded as the source of a bibliographic variant in the third author's name; also our source for two of the competing motivation papers, neither of which we obtained. resolve.cambridge.org
- Publisher record for a transcontinental ultimatum game study, carrying a reference list which renders the 1982 first author as Güth W.R., adding a middle initial not present elsewhere, and the third author as Schwarz B. without the final letter; and confirming Oosterbeek, H., Sloof, R., and van de Kuilen, G. (2004), Experimental Economics, 7(2), 171–188; Kagel, J. H., Kim, C., and Moser, D. (1996), Fairness in ultimatum games with asymmetric information and asymmetric payoffs, Games and Economic Behavior, 13(1), 100–110; and Rabin, M. (1993), Incorporating fairness into game theory and economics. Note: a publisher's reference list. Recorded as the source of two further name variants; also used to confirm the meta-analysis citation independently. link.springer.com
This article discusses research on bargaining and is not negotiation, legal or commercial advice. The 2004 meta-analysis was obtained as an abstract only, from six independent sources, plus working paper passages; no magnitudes for its regional differences are reported. The 1982 paper and all cross-cultural studies were not obtained. All arithmetic is the authors' own: the leverage calculation is exact, and the optimal-offer, gap and flat-maximum sections use an entirely invented rejection curve whose parameters no source supplies. The authors' first draft of the optimal-offer range was wrong and is corrected in the body. The standard experiment forbids counteroffers, which is true of almost no business negotiation.