The tipping point: a mathematical model for the profit-driven abandonment of restaurant tipping
Sara M. Clifton, Eileen Herbers, Jack Chen, Daniel M. Abrams
TThe tipping point: a mathematical model for the profit-drivenabandonment of restaurant tipping
Sara M. Clifton ∗ Department of Mathematics, University of Illinoisat Urbana-Champaign, Urbana, IL 61801, USA
Eileen Herbers and Jack Chen
Department of Engineering Sciences and Applied Mathematics,Northwestern University, Evanston, IL 60208, USA
Daniel M. Abrams
Department of Engineering Sciences and Applied Mathematics,Northwestern University, Evanston, IL 60208, USANorthwestern Institute for Complex Systems,Northwestern University, Evanston, IL 60208, USA andDepartment of Physics and Astronomy,Northwestern University, Evanston, IL 60208, USA (Dated: December 18, 2017)
Abstract
The custom of voluntarily tipping for services rendered has gone in and out of fashion in Amer-ica since its introduction in the 19th century. Restaurant owners that ban tipping in their estab-lishments often claim that social justice drives their decisions, but we show that rational profit-maximization may also justify the decisions. Here, we propose a conceptual model of restaurantcompetition for staff and customers, and we show that there exists a critical conventional tip rateat which restaurant owners should eliminate tipping to maximize profit. Because the conventionaltip rate has been increasing steadily for the last several decades, our model suggests that restaurantowners may abandon tipping en masse when that critical tip rate is reached. ∗ E-mail me at: [email protected] a r X i v : . [ q -f i n . E C ] D ec . INTRODUCTION Tipping for restaurant service has gone in and out of fashion in America since its in-troduction from Europe in the 19th century [1]. Tipping has always been a controversialsocial convention, for both scholars and the public. The practice has been consistently tiedto the worst of human nature: racism, sexism, and classism [2–5]. At many points in time,tipping has been considered downright anti-democratic [6]. Yet the practice persists becausethe vast majority of Americans prefer to choose how much gratuity they leave after a meal[7, 8].Economists have traditionally struggled to explain the practice of tipping in terms ofrational costs and benefits because a rational economic agent would not incur a monetarycost that provides no present or future benefit [9]. Sociologists and psychologists haveappealed to negative feelings of guilt, embarrassment, or anxiety to explain why peopleconform to the social convention of tipping [10–12], while others have appealed to altruisticfeelings of generosity and empathy [13–15].While much scholarly attention has been paid to the consumers who tip and employeeswho receive tips (e.g. [16–19]), relatively little has been paid to restaurant owners whoemploy tipped staff. Theoretical models of tipping often see restaurant owners as inefficientjudges of service quality [20]. The natural conclusion to this rationale is that restaurantowners should allow tips in their restaurants in order to efficiently evaluate the quality oftheir wait staff [21]. This conclusion breaks down when we consider the real-world factorsthat influence restaurant owners’ decisions, such as remaining profitable while also adheringto minimum wage restrictions, retaining talented staff, and meeting customer expectationsfor food and service quality [22]. By accounting for these factors and appealing only torational profit-maximization motivations, we show that restaurants owners should play amore active role in determining tip rates in their restaurants.As the conventional tip rate gradually increases in the US (see Fig 1), waiters’ take homepay steadily increases, while back-of-house employees’ pay remains stagnant [23]. Despite thelow federal minimum wage for tipped workers ($2.13 as of 2017 [24]), waiters consistentlyearn more than cooks [25]. As the wage disparity increases, talented cooks may defectto restaurants where profits are shared more equitably among staff, and talented waitersmay defect to restaurants with higher tips. A rational restaurant owner interested only inmaximizing profit might take control of the tip rate in his/her restaurant in order to retainthe most talented front-of-house (tipped) and back-of-house (untipped) staff. We show in aconceptual model of two competing restaurants that a critical tipping rate exists at whicha rational restaurant owner will abandon tipping to maximize profit.
II. METHODSA. Model derivation
As a simple conceptual model, consider two restaurants competing for diners, waiters,and cooks. For notational ease, we will focus on one restaurant ( our restaurant , Restaurant1). We assume all diners, waiters, and cooks intend to eat or work, respectively, at eitherour restaurant or the competing restaurant. Following a dynamical systems approach byAbrams et al. for modeling generic social group competition [27], people will transitionbetween restaurants at a rate proportional to the relative utility of being at each restaurant2
980 1990 2000 2010 202014151617181920 year a v e r age t i p r a t e ( % ) FIG. 1. Average reported tip rate in American restaurants over time, according to NPD Group(1982-84) and Zagat annual surveys (1989-present) [26]. Note that both surveys are aimed at dinerswho patronize full-service midscale and upscale restaurants. (either as a customer or an employee). This general model assumes that the popularity ofa social group also influences the transition rates, but for our purpose of modeling purelyrational behavior, we focus only on utility as a driving force. Our simplification reduces themodel by Abrams et al. to d x d t = y P yx ( u x ) − x P xy ( u x ) , (1)where x is the fraction of people in group X , P yx is the probability per unit time of tran-sitioning from group Y to group X , u x is the utility of group X , and transition rates aresymmetric under exchange of x and y . In our case, X is our restaurant and Y is the com-peting restaurant. Alternatively, the “competing restaurant” could be viewed as a reservoirof all other dining options (including home), but that extension is left for future work.
1. Dynamics of cooks
Suppose our system has a number N C of cooks who must choose between our restau-rant and the competing restaurant. Many factors influence a cook’s decision to work at aparticular restaurant, but we model cooks primarily concerned with pay. Because cooks donot receive tips, cooks choose where to work based only on the relative wage at the tworestaurants. Assuming for simplicity that the transition probabilities are linear in relativebase cook pay, the change in the number of cooks ˜ C at our restaurant is τ C d ˜ C d t = ( N C − ˜ C ) b C b C + b C (cid:124) (cid:123)(cid:122) (cid:125) switch fromcompetitor to us − ˜ C b C b C + b C (cid:124) (cid:123)(cid:122) (cid:125) switch fromus to competitor , (2)3here τ C sets the transition time scale for cooks, and b C and b C are the hourly base cookpay at our restaurant and the other restaurant, respectively. For example, if both restaurantsoffer the same base pay ( b C = b C ), then eventually half the cooks will be at our restaurant,and the other half will go to the competitor. Tilde notation will be removed later when themodel is normalized.
2. Dynamics of waiters
Like the cooks, the waiters in our system are primarily concerned with pay. Becausewaiters receive both hourly wages and gratuity, the transition rate between restaurantsdepends on the relative hourly take home (total) pay at the two restaurants. The hourlygratuity at our restaurant is g = m ˜ DT ˜ W , (3)where m is the hourly menu price, ˜ D is the number of diners, T is the tip rate, and ˜ W is the number of waiters that must split the total tips. The gratuity g at the competingrestaurant is similarly defined.The hourly take home pay at our restaurant is then b W + g , where b W is the hourlybase waiter pay at our restaurant. The change in the number of waiters ˜ W at our restaurantis then τ W d ˜ W d t = ( N W − ˜ W ) b W + g b W + g + b W + g (cid:124) (cid:123)(cid:122) (cid:125) switch fromcompetitor to us − ˜ W b W + g b W + g + b W + g (cid:124) (cid:123)(cid:122) (cid:125) switch fromus to competitor , (4)where τ W sets the transition time scale for waiters, and N W is the number of waiters in thesystem.
3. Dynamics of diners
Assuming all diners intend to eat at a restaurant, they must chose between our restaurantand our competitor. Many factors influence a person’s decision to eat at a particular restau-rant, but we will focus on food and service quality versus menu cost. There are also manyways to measure food and service quality [28–30], but we will use the number of cooks andwaiters who choose to work at our restaurant as a basic proxy. For instance, if our restaurantattracts more waiters, then diners will receive more personal attention and perceived servicequality will increase. Suppose for simplicity that the quality q of the meal and service atour restaurant is a linear combination of the number of cooks ˜ C and waiters ˜ W working atour restaurant: q = α W ˜ W + α C ˜ C, (5)where α W and α C are the weights placed on service and food, respectively, when evaluatingour restaurant. The quality q of the competing restaurant is defined similarly. Note thatthis proxy for restaurant quality will only be an acceptable approximation if both restaurants4re not grossly over- or under-staffed. We explore two alternative formulations for restaurantquality in the supplementary online material, and we find similar qualitative results.We define the value v of our restaurant as the quality q over the menu cost (includingtips): v = α W ˜ W + α C ˜ Cm (1 + T ) , (6)where m is the hourly menu cost and T is the tip rate at our restaurant. The value v ofthe other restaurant is defined similarly.A rational diner chooses a restaurant based on the perceived relative value of each restau-rant. The change in the number of diners ˜ D at our restaurant is then τ D d ˜ D d t = ( N D − ˜ D ) v v + v (cid:124) (cid:123)(cid:122) (cid:125) switch fromcompetitor to us − ˜ D v v + v (cid:124) (cid:123)(cid:122) (cid:125) switch fromus to competitor , (7)where τ D sets the transition time scale for diners, and N D is the number of diners in thesystem.
4. Profitability
Given the flow of employees and customers to and from our restaurant, a rational restau-rant owner will maximize hourly profit˜ P = m ˜ D (cid:124) (cid:123)(cid:122) (cid:125) revenue − b W ˜ W (cid:124) (cid:123)(cid:122) (cid:125) waiter pay − b C ˜ C (cid:124) (cid:123)(cid:122) (cid:125) cook pay . (8)We ignore fixed costs because we are only concerned with maximizing profitability, notabsolute profits. B. Normalized model
We now normalize and nondimensionalize the system (2)-(7) to reduce the number ofparameters. We make the following substitutions D = ˜ DN D , W = ˜ WN W , C = ˜ CN C (9) r = α C α W , r DW = N D N W , r CW = N C N W , (10)so that D , W , and C are the fraction diners, waiters and cooks at our restaurant, r isthe ratio of food to service importance for customers, and r DW and r CW are the ratios ofdiners and cooks to waiters, respectively. We also rescale time such that τ C = τ W = τ D = 1.Naturally, the transition rates may vary for diners, waiters, and cooks; customers may switchdining locations more rapidly than employees switch jobs. However, we are only interestedin equilibrium states, so we ignore this detail.5hen the fraction of diners at our restaurant follows the dynamicsd D d t = (1 − D ) v v + v − D v v + v (11) v = W + r r CW Cm (1 + T ) , v = (1 − W ) + r r CW (1 − C ) m (1 + T ) . (12)The fraction of waiters at our restaurant follows the dynamicsd W d t = (1 − W ) b W + g b W + g + b W + g − W b W + g b W + g + b W + g (13) g = m r DW DT W , g = m r DW (1 − D ) T W . (14)Finally, the fraction of cooks at our restaurant follows the dynamicsd C d t = (1 − C ) b C b C + b C − C b C b C + b C . (15)All variables and parameters are described in Table I. Variable Meaning Units Range Baseline D fraction of diners at our restaurant – [0, 1] – W fraction of waiters at our restaurant – [0, 1] – C fraction of cooks at our restaurant – [0, 1] – t time (dimensionless) – [0, ∞ ) – r relative importance of food quality versus service quality,typically a value exceeding one – [4, 20] 12 § r CW ratio of total cooks to waiters in the system – [0.5, 2] 1 r DW ratio of total diners to waiters in the system – [1, 20] 12 m average menu cost per hour at our restaurant $/hr [5, 20] 10 b W waiters’ base pay per hour at our restaurant $/hr [2.13*, 25] 5.00 † b C cooks’ base pay per hour at our restaurant $/hr [7.25*, 25] 10.40 † T average tip rate at our restaurant, determined by eithersocial convention or mandated by restaurant owner – [0.01, 0.5] 0.19 ‡ v meal value perceived by customers at our restaurant 1/$ – – g gratuity per hour at our restaurant $/hr – –TABLE I. Description of model variables and parameters for our restaurant, Restaurant 1. Thecompeting restaurant (Restaurant 2) has similarly defined parameter values subscripted with 2.We present a range of plausible values for each parameter and a baseline value for midscale andupscale restaurants like those reviewed by Zagat. ( § crude estimate based on customer surveys[28]; *federal minimum wage as of 2017 [24]; † average waiter and cook pay as of 2015 [31]; ‡ averageself-reported tip rate as of 2016 [26]; other baseline values are guesses based on author experience). With this change of variables, hourly profit ˜ P becomes the hourly profit per waiter inthe system P = m r DW D − b W W − b C r CW C. (16)6 II. RESULTSA. Numerical exploration
Numerical integration suggests that one stable steady state solution exists for each set ofparameters regardless of the initial condition, so long as the initial condition is physicallymeaningful. Because cooks only switch restaurants in response to base pay (constant pa-rameter), the distribution of cooks equilibrates first. Diners and waiters respond to everyoneelse in the system, so the distribution of diners and waiters equilibrates later.For otherwise identical restaurants, small changes in restaurant policy (like staff pay,menu prices, and tip rates) will have an effect on the entire restaurant ecosystem. Loweringthe tip rate at our restaurant will cause waiters will leave our restaurant because they getpaid less, but diners will prefer our restaurant because they pay less (see Fig 2a).If the menu price at our restaurant is lower than the competitor, then diners will flockto our restaurant because they pay less, and waiters will temporarily leave our restaurantbecause lower menu prices lead to lower tips. However, after our restaurant has a large shareof diners, waiters return because the density of diners balances the lower menu prices (seeFig 2b).If we pay our cooks less than our competitor, then cooks will leave our restaurant becausethey get paid less; as food quality decreases, diners will leave our restaurant, and then waiterswill leave our restaurant as their hourly tips decrease (see Fig 2c).As a final example, if we pay our cooks more but pay our waiters less to compensate,cooks will flock to our restaurant followed by diners; waiters will temporarily leave becausethey are paid lower wages, but eventually they will come back as diners flood our restaurant(see Fig 2d).
B. Equilibrium stability analysis
Fixed point analysis shows that four steady states exist. Only one fixed point is mean-ingful (i.e., D ∗ , W ∗ , C ∗ ∈ [0 , C ∗ = b C / ( b C + b C ). Thesteady states for waiters and diners have closed forms but are too long to include. For allreasonable parameter values (listed in Table I), the eigenvalues of the Jacobian evaluated atthe fixed point are real and negative. This implies that the equilibrium is a stable sink. SeeFig S5. C. Equilibrium sensitivity analysis
Global sensitivity and uncertainty analysis using Latin Hypercube Sampling (LHS) ofparameter space and Partial Rank Correlation Coefficients (PRCC) [32] reveal that equilib-rium distributions of diners and waiters depend significantly ( p < . .480.50.51 time (unitless) f r a c t i on o f d i ne r s , w a i t e r s , and c oo ks a t ou r r e s t au r an t dinerswaiterscooks a bdc FIG. 2. (Color online) Numerical simulation of system (11)-(15) with two nearly identical compet-ing restaurants. (a)
If the tip rate at our restaurant is lower than the competitor, then waiters willleave, but diners will prefer our restaurant ( T = 0 . (b) If the menu price at our restaurantis lower than the competitor, then diners will flock to our restaurant, and waiters will temporarilyleave. However, after our restaurant has a large share of diners, waiters will return ( m = 15). (c) If we pay our cooks less, then cooks followed by diners followed by waiters will leave ( b c = 12). (d) If we pay our cooks more but pay our waiters less to compensate, cooks will flock to our restaurantfollowed by diners; waiters will temporarily leave because they are paid lower wages, but eventuallythey will come back as diners flood our restaurant ( b w = 10 , b c = 15). Unless otherwise noted, m = m = 10 , T = T = 0 . , b w = b w = 5 , b c = b c = 10 , r = 12. IV. DISCUSSIONA. Tip abandonment threshold
Suppose our restaurant is attempting to maximize hourly profit (16) at equilibrium. Weassume our restaurant is competing with a typical American restaurant that is not makingdynamic changes to staff pay, menu prices, or tipping policies. Given the choices the otherrestaurant has made, our restaurant can choose base pay for cooks and waiters (within legallimits) and a gratuity policy. Both restaurants maintain identical menu prices to ensure therestaurants are true competitors; fine dining establishments do not typically compete withcasual restaurants.If the competing restaurant allows the conventional tipping rate, then there exists acritical tip rate threshold T c at which a rational restaurant owner would forbid tipping to8aximize profit. Fig 3a shows the conventional tip rate at which a hypothetical restaurantshould switch from allowing the conventional tip to abandoning tipping in their establish-ment. Though the critical tip rate depends on the entire restaurant ecosystem, numericalexploration indicates that the trade off between meal cost and restaurant quality, as per-ceived by diners, is the primary driver of the critical tip rate (see Fig 3b-f). Assumingthe conventional tip rate continues to increase in the US, we predict that restaurants willeventually forbid tipping when it become more profitable to do so.Global sensitivity and uncertainty analysis shows that this critical tipping threshold de-pends significantly on the menu price shared by both restaurants, the ratio of customers towaiters and cooks to waiters, and the ratio of food quality to service quality in the eyes of thecustomer (see Fig 4). Note that the parameters that significantly influence the critical tiprate T c describe the type or “class” of restaurant system we are considering. For instance,fine dining restaurants maintain a low diner to waiter ratio r DW and high menu prices m .It is also likely that diners at fine dining establishments place more value on service than atcasual restaurants, decreasing r .Local sensitivity analysis about ‘typical’ American restaurant parameters suggests thatincreased menu price, increased service importance, increased diner-to-waiter ratio, andincreased waiter-to-cook ratio all increase the critical tipping rate (see Fig 5). Becauseno type or class of restaurant increases all these parameters, we cannot say with certaintythat a certain type of restaurant should abandon tipping before another. However, thethree strongest correlated parameters ( r, r CW , m ) support the prediction that casual diningrestaurants should be the first to abandon tipping, and fine dining establishments should bethe last to abandon tipping.This prediction is consistent with tipping practices in the most casual restaurants: cus-tomers in fast food restaurants and counter-service establishments are not typically expectedto leave tips. Among restaurants that expect patrons to leave tips, the prediction is surpris-ing because the most vocal advocates for eliminating tipping in America have been ownersof upscale restaurants. However, fine dining restaurant owners cite social justice as the pri-mary motive for eliminating tipping in their establishments [33]. This claim is consistentwith our prediction because many restauranteurs have been forced to reinstate tipping intheir restaurants in order to remain profitable [34]. B. Limitations
As a conceptual model, system (11)-(15) cannot offer quantitative predictions with con-fidence. One limitation of this model is the lack of competition among many restaurants oreating at home, though this could be addressed by considering the “competing restaurant”as a pool of competition. Additionally, the model assumes that the benefit of more employ-ees does not have diminishing returns. More realistically, restaurant food or service will onlybenefit from more employees up to a certain point; after the restaurant is fully staffed, moreemployees will be a waste of money and may even impede service.Our model also ignores both the federal law that requires restaurant owners to supplementtipped worker wages if their hourly tips do not exceed the federal minimum wage [35] andmany state laws that impose larger minimum wages for tipped employees [24]. We also do notprovide a mechanism by which the conventional tip rate increases and merely assume thatthe increasing trend will continue; however, the increasing trend is supported by theoreticaleconomic models [9]. Finally and most importantly, this model assumes that humans behave9 ip rate at competing restaurant r e s t au r an t p r o f i t i b ili t y ( a r b i t r a r y un i t s ) T = T T = allow tippingforbid tipping T C tip rate d i ne r v a l ue r a t i o s ( v / v ) tip rate c oo k pa y r a t i o s ( b C / b C ) w a i t e r t o t a l pa y r a t i o s ( b W + g1 ) / ( b W + g2 ) tip rate r e s t au r an t qua li t y r a t i o s ( q1 / q2 ) tip rate e ff e c t i v e m enu p r i c e r a t i o s ( m / m ) tip rate a bcd e f FIG. 3. (Color online) Example of critical tip rate threshold. (a)
For conventional tip rates belowsome critical threshold T c , a rational restaurant owner would allow diners to leave gratuity to max-imize profitability (black curve). Beyond that critical threshold, a rational restaurant owner woulddisallow tipping in their restaurant (red dashed curve). Both curves assume that the restaurantowner selects staff pay (within legal limits) to maximize profit. For this hypothetical restaurantecosystem, (b) both optimal cook pay and (c) total waiter pay (optimal wage plus tips) drop if weeliminate tipping. Though staff leave our restaurant in response to a no-tip policy, (d) the dropin perceived quality balances (e) the effective menu price (menu cost plus tips) near the criticaltip rate. (f ) The trade off between quality and price, as perceived by diners, drives the criticaltip rate. For this example, m = m = 10 , r = 4 , b W = 10 , b C = 25 , r DW = 10 , r CW = 1, theminimum wage for tipped workers is 2 .
13, and the minimum wage for untipped workers is 7 . rationally when spending or earning money, a false assumption common among economicmodels [36]. Restaurant owners may choose to abandon or maintain tipping regardless ofprofit, citing economically irrational reasons or responding to irrational customer feelings.In spite of these limitations, the qualitative prediction that a critical tipping thresholdexists at which restaurant owners may abandon tipping is supported by previous trends10 rcw rdw m bw2bc2−11 P RCC ( t i pp i ng t h r e s ho l d T c ) parameter *** *** *** *** FIG. 4. Global sensitivity and uncertainty analysis for tipping threshold T c . The Partial RankCorrelation Coefficient (PRCC) between model parameters and the tipping threshold T c are show.Asterisks indicate that the correlation is significant (*** p < . , N = 100 samples). Note thatwe use PRCC because numerical tests suggest that the relationship between parameters and T c ismonotonic. both in America and internationally. Tipping has gone in and out of fashion around theworld, and though customers normally drive the introduction (or reintroduction) of thetrend, restaurant owners or governments typically end the practice [37]. V. CONCLUSION
The conceptual model presented here takes a new direction towards understanding thecomplex service industry. The oscillating popularity of tipping has previously been at-tributed to social contagion and irrational responses to classism. Using a new approachto modeling the social convention of tipping, we show that rational decisions to maximizeprofit may drive the cycle of the tipping trend. We predict that there exists a critical tiprate threshold at which restaurant owners would be wise to eliminate tipping in their estab-lishments. Furthermore, we expect that casual restaurants should be the first to abandontipping, and fine dining restaurants should be the last.The simplicity of the model does not allow for quantitative predictions, such as whentipping will go out of fashion in the US or what the threshold tip rate will be. However,the model serves as a base for more sophisticated models and could direct economic datacollection to better answer quantitative questions. This effort would be important not onlyto restaurant owners, but also to economists, sociologists, policy makers, and all people whoplay a role in or interact with the service industry.11 t i pp i ng t h r e s ho l d ( T c ) t i pp i ng t h r e s ho l d ( T c ) ratio of cooks to waiters a bc d FIG. 5. Local sensitivity analysis for tipping threshold T c . (a) Holding all else constant, highermenu price implies a larger tip threshold. This indicates that fine dining restaurants should be thelast the abandon tipping if all other parameters are the same. (b)
As diners place more relativeimportance on food than service, the critical tip rate decreases. Because customers at fine diningestablishments likely place more value on service, we again expect that fine dining restaurantswill be the last to abandon tipping. (c)
In contrast, as the ratio of diners to waiters increases,the critical tipping rate increases. All else held constant, this would imply that casual diningestablishments would drop tipping last. (d)
As the ratio of cooks to waiters increases, the criticaltip rate decreases. Though it is difficult to know this ratio for full service restaurants, it makessense that the large cook-to-waiter ratio seen at counter service restaurants implies little to notipping. For this example, m = m = 10 , r = 12 , b W = 5 , b C = 10 , r DW = 12 , r CW = 0 .
5, unlessotherwise noted on the independent axis.
VI. SUPPLEMENTARY MATERIAL
See supplementary material for additional discussion and figures.
VII. ACKNOWLEDGMENTS
The authors thank Michael Lynn for sharing both data and ideas. This work was fundedin part by the National Science Foundation Graduate Research Fellowship No. 1324585.Funding was also provided by the Northwestern University Undergraduate Research Assis-tant Program. The funders had no role in study design, data collection and analysis, decision12o publish, or preparation of the manuscript.
VIII. DATA AVAILABILITY
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1. OPTIMAL WAITER BASE PAY
The profit optimization algorithm selects the most profitable staff pay rate for every tiprate scenario, so restaurant owners that abandon tipping must pay waiters a higher wageto compensate for the lost tips. Fig S1 shows that the relative optimal base pay when weforbid tipping must steadily increase as the tip rate increases. Tc conventional tip rate w a i t e r ba s e pa y r a t i o s ( b W / b W ) allow tippingforbid tipping FIG. S1. Illustration of optimal waiter base pay under different tipping protocols. When ourrestaurant chooses to forbid tipping, the optimal waiter base pay rate increases steadily with theconventional tip rate. In other words, restaurant owners must compensate for lost tips. For thisexample, m = m = 10 , r = 4 , b W = 10 , b C = 25 , r DW = 10 , r CW = 1, the minimum wage fortipped workers is 2 .
13, and the minimum wage for untipped workers is 7 .
25. Note that the base payratio is low when tipping is allowed because waiters at the other restaurant are paid $10 an hourno matter what, and the optimization algorithm tells us to pay waiters minimum wage ($2 . As the tip rate increases, waiters receive relatively more compensation from tips thanfrom base pay (see Fig S1). This is expected, though it is difficult to know if our ratios arerealistic because tips, often paid in cash, may be under-reported.
S2. ALTERNATIVE INTERPRETATIONS OF RESTAURANT QUALITY
Though we have framed the model as counting the literal number of waiters and cooksin the restaurant, the model could be reinterpreted as measuring the total waiter and cookquality. Because the number of cooks and waiters in our restaurant is actually the numberwho wish to work in our restaurant, one could argue that W and C are indicators of thequality of staff we’ll have when we hire the appropriate number of waiters and cooks.Without any adjustment to the model, C could be interpreted as the fraction of cookswho wish to work in our restaurant, and b C C (total pay for all cooks) could be the quality15 Tc conventional tip rate w a i t e r ba s e pa y t o t o t a l pa y r a t i o ( b W / ( b W + g1 )) allow tippingforbid tipping FIG. S2. Illustration of base pay as a fraction of total pay under different tipping protocols. Whenour restaurant allows tipping, less and less waiter pay comes from wages as the conventional tiprate increases. For this example, m = m = 10 , r = 4 , b W = 10 , b C = 25 , r DW = 10 , r CW = 1,the minimum wage for tipped workers is 2 .
13, and the minimum wage for untipped workers is 7 . of the cooks actually hired. We simply assume that we hire the best cooks from the poolof applicants and pay them appropriately. For instance, if ten cooks want to work at ourrestaurant, then we could hire all ten at $10 an hour. Equivalently, we could hire only themost qualified five cooks and pay them $20 an hour.We can apply the same logic to waiters, but the model will need to be adjusted slightly.As is, gratuities are split among all waiters, and a new model would need to account for theactual number of waiters working in the restaurant. For instance, if we hire half as manywaiters that are twice as talented and pay them twice the salary, our profit function wouldnot change. However, each waiter would receive double the tips because the gratuities wouldbe split among half as many waiters. S3. ALTERNATIVE FORMULATIONS FOR RESTAURANT QUALITY
The most difficult modeling task for this restaurant ecosystem is formulating a simple, butrealistic, restaurant quality measure. The original model uses the weighted sum of waitersand cooks at our restaurant as a proxy for restaurant quality. Alternatively, a weighted sumof waiter and cook pay could be a proxy for restaurant quality: q = α W ( b W + g ) + α C b C (17)With this new definition of restaurant quality, the critical tip threshold remains (see FigS3). 16 conventional tip rate r e s t au r an t p r o f i t i b ili t y allow tippingforbid tipping FIG. S3. Example of critical tip rate threshold with pay-dependent quality measure (17). Forconventional tip rates below some critical threshold (blue dashed line), a rational restaurant ownerwould allow diners to leave gratuity to maximize profitability (black curve). Beyond that criticalthreshold, a rational restaurant owner would disallow tipping in their restaurant (red dashed curve).Both curves assume that the restaurant owner selects staff pay (within legal limits) to maximizeprofit. For this example, m = m = 10 , r = 2 , b W = 10 , b C = 25 , r DW = 10 , r CW = 1, theminimum wage for tipped workers is 2 .
13, and the minimum wage for untipped workers is 7 . Another alternative proxy for restaurant quality could depend on the product of staff payand number of staff: q = α W ˜ W ( b W + g ) + α C ˜ Cb C (18)With this new definition of restaurant quality, the critical tip threshold again remains (seeFig S4). This suggests that the qualitative prediction that a critical tip rate exists is robustto the exact model formulation for restaurant quality.17 conventional tip rate r e s t au r an t p r o f i t i b ili t y allow tippingforbid tipping FIG. S4. Example of critical tip rate threshold with pay- and staff-dependent quality measure (18).For conventional tip rates below some critical threshold (blue dashed line), a rational restaurantowner would allow diners to leave gratuity to maximize profitability (black curve). Beyond thatcritical threshold, a rational restaurant owner would disallow tipping in their restaurant (red dashedcurve). Both curves assume that the restaurant owner selects staff pay (within legal limits) tomaximize profit. For this example, m = m = 10 , r = 4 , b W = 10 , b C = 25 , r DW = 10 , r CW = 1,the minimum wage for tipped workers is 2 .
13, and the minimum wage for untipped workers is 7 .
4. ADDITIONAL FIGURES f r a c t i o n w a i t e r s FIG. S5. Phase portrait of two identical restaurants with differing tip rates. Our restaurant(shown) enforces an automatic gratuity of T = 0 .
15, and the competing restaurant allows theconventional tip rate of T = 0 .
2. The steady state is ( D ∗ , W ∗ , C ∗ ) = (0 . , . , . D/ d t = 0 (blue) and d W/ d t = 0 (red) are superimposed. For this example, m = m = 10 , r =12 , b W = b W = 5 , b C = b C = 10 , r DW = 1 , r CW = 1. rcw rdw m T1 T2 bc1 bc2 bw1 bw2 −11 −11 Parameter Parameter P RCC ( d i ne r equ ili b r i u m ) P RCC ( w a i t e r equ ili b r i u m ) *** *** *** *** *** *** *** *** *** *** r rcw rdw m T1 T2 bc1 bc2 bw1 bw2 a b FIG. S6. Global sensitivity and uncertainty analysis for equilibrium state. (a)
Partial Rank Corre-lation Coefficient (PRCC) between model parameters and diner equilibrium. (b)
PRCC betweenmodel parameters and waiter equilibrium. Asterisks indicate that the correlation is significant(*** p < . , N = 100 samples). Note that we use PRCC because numerical tests suggest thatthe relationships between parameters and equilibria are monotonic.= 100 samples). Note that we use PRCC because numerical tests suggest thatthe relationships between parameters and equilibria are monotonic.