February 27, 2005
Beyond Homo economicus
[Note: Much of this post is based on Chapter 11 of Game Theory Evolving: A Problem-centered Introduction for Modeling Strategic Interaction by Herb Gintis. For those who find this post interesting, buying this book would be a good place to start. I have also included some of the citations by Gintis in this post so that the interested reader will know where to look for further information.]
The standard model for human behavior in economics is that the individual maximizes his utility subject to a set of constraints. This view of human behavior is described as Homo economicus. But there are problems with this version of human behavior. For example, in public goods game experiments there often starts out with a high degree of cooperation that starts to decay later in the game. Also, there are problems with the way people make decisions under uncertainty. In this last case it is quite often the case that people don't make the right choice given that the model of human behavior is expected utility maximization.1 Of course this does not mean that Homo economicus is everywhere and anywhere a bad model for human behavior.
Homo economicus has provided a wealth of results when there are well defined markest (double continuous auctions) as well as oligopoly models (Davis and Holt 1993, Kachelmaier and Shehata 1992). So Home economicus is sometimes useful, but at other times it is not. For example, when there is not complete information and there are opportunities for strategic behavior many of the predictions derived from the Homo economicus fail to materialize. Sometimes, the agents will retaliate against defectors even when retaliation comes at a cost. Other time agents will take actions that will reduce their overall welfare but enhance equality. Further, it should be understood that these actions are not necessarily irrational in an over all sense, they are simply irrational given the Homo economicus hypothesis for human behavior. So, the standard assumption/hypothesis used in much of economics is one that needs re-working, or at least one should be careful when invoking that model.
Homo equalis
One new model for human behavior is Homo equalis. Homo equalis actually has preferences for equality and this will mean the behavior for Homo equalis will deviate from that of Homo economicus.
Homo equalis exhibits a weak urge to reduce inequality when on top, and a strong urge to reduce inequality when on the bottom.(Gintis 2000)
The orgin of Homo equalis is from the anthorpological literature (Gintis 2000). In small hunter/gatherer groups there are no centralized governments, and individuals that deviate from voluntary cooperation are often banished from the group or killed. As a result widespread sharing of resources (shelter, food, etc.) is common.
In terms of experimental game theory results such as the ultimatum game depending on the exact "shape" of the utility function of Homo equalis2 the offers that are made well vary, but will tend to be well above the sub-game perfect strategy of offering a split of ($9,$1) which the Homo economicus assumption predicts. The modal offer from experimental evidence is around 50% (i.e. an offer of ($5,$5)) with respondents rejecting offers below 30% (i.e., rejecting offers below ($7,$3)) (Camerer and Thaler 1995, Guth and Tietz 1990).
Another game where the results using Homo equalis measure up better than Homo economicus is the public goods game. In this game the players can put a dollar, form their private account, into the public pot. At the end of the round the experimenter will deposit $.5 for each dollar in the public account into the players accounts. The process is repeated ten times. Perfect cooperation will result in $50 in each players account (which the player can keep). If one player were to defect he'd have $55 and the rest of the players would have $45. If everybody defects each player keeps $10. The sub-game perfect equilibrium (and the predicted outcome with the Homo economicus hypothesis) for this game is for nobody to contribute to the public pot and for each player to get $10. The exeprimental evidence on the other hand have very different results. Further, with Homo equalis, again depending on the shape of the utility function, there can be substantial cooperation.
Homo reciprocans
I have blogged previously about Homo reciprocans, but there is more than that short post covers. The problem with Homo equalis is that while it does do a decent job of reproducing what is seen in game theory experiments it does not address the issue of what agents want to see in terms of the intentions of others. In other words, Homo equalis does say much about how "cheaters" are viewed. Typically cheaters are viewed poorly, and often times people will seek ways to retaliate against "cheaters" even at a cost to themselves. Homo reciprocans allows for this kind of behavior.
One experiment that suggests that people care about the intentions of other players is where the proposer in the ultimatum game is a computer and the responders know this. In this case, the number of rejections fall significantly (Blount, 1995). the idea here being that the responder's rejections will have little punishment effect on a computer. The reciprocal nature of play can also be seen in much earlier research such as Robert Axelrod's competitions on game theory. The tit-for-tat startegy has play dependent on reciprocity.
Of course, the first objection that could be raised is that "co-operation" in the above sense is nothing more than "enlightened self interest". That is, in repeated game contexts with discount rates that are sufficiently low cooperative outcomes will result even with the assumption of Homo economicus. However, the problem is that the level of reciprocity that is observed in experimental games could be called strong reciprocity in that it applies in situations where the above doesn't have to hold. That is, there is a tendency to cooperation and punish even when there are costs to doing so. Further, this kind of behavior can be seen in things like one-shot games where the above arguments of "enlightened self-interest" in conjunction with Homo economicus should not hold.
Homo reciprocans offers up a story for such observations as rejecting proposals in the Ultimatum game. Note that with Homo equalis all of the results center around the proposals with regards to the Ultimatum game and not the rejections. With Homo reciprocans we actually can say something about both the proposer and the respondent. Further, Homo economicus behavior is an extreme case (this arises when the individual cares nothing about fairness of others behavior and kindness towards others).
Homo parochius
Homo parochius divides the world in two groups insiders and outsiders. This division can be along racial lines, ethnicity, language, nationality, and even random factors3. While the behavior of Homo parochius is generally condemned by society at large such behavior can often be elicited in experimental games suggesting that even with "moral training" against such behavior it is still present in many people. Some examples one form that is considered innocuous is hometown favortism. Most people in a community with a sports team will likely favor other individuals that support the hometeam. Homo economicus, Homo equalish, and Homo reciprocans all fail to capture this kind of behavior.
The traditional model/assumptions used in economics about the behavior of individual actors is clearly not sufficient to cover all areas where economic analysis can be applied. As such these new models offer additional "tools" to the researcher. One of the past objections to such models is that they are "ad-hoc". That is the models are constructed because this is what we observe in individual/group behavior. The problem with this approach is exemplified in the Phillips curve inflation/unemployment trade off. It is a purely empirical result that could vanish if policy is based on the model. However, as I noted in this post evolutionary game theory has provided a solid theoretical basis for why such models for individual behavior. Caring about reciprocity, equality and even insiders vs. outsiders could provide evolutionary advantages. Thus, this type of thinking becomes dominant while purely selfish (Homo economicus) loses ground or doesn't develop until later.
Note that this does not make Homo economicus worthless or unimportant. In the public goods games for example a percentage of the players do play according to Homo economicus and this could be one reason why the cooperation in early stages deteriorates to the point where in later stages everybody is playing according to Homo economicus.
While the above address some of the problems noted in the first paragraph that face economic theory, it does not address them all. Further, there is still resistance in the economics profession to these ideas. However, as the evidence continues to mount and with Daniel Kahneman's Nobel award, the momentum, in my view, is with those who favor incorporating these new ideas into economics.
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1A good example of this is the Monty Hall problem. The Monty Hall problem is as follows: You have to pick between three doors. Behind one door is a car, behind the others is nothing. Now, in the first round Monty offers you a choice between the three doors. At this point most people provide the right answer that their chances of winning is 1/3rd. After making your selection Monty does the following: he opens one of the two remaining doors that does not have the car behind it. Upon doing this Monty offers you the option of switching. At this point most people incorrectly asses their probability of winning as being 1/2 and as a result do not switch to the other remaining door. However, this is the wrong strategy. Even though Monty has opened the door with nothing behind it this does nothing to the probability that the car is behind your initial pick. Switching always provides a probability of 2/3rds for winning and is the dominanat strategy.
Davis, D. D. and C. A. Holt 1993. Experimental Economics. Princeton, NJ: Princeton University Press.
Kachelmaier, S. J. and M. Shehata 1992. "Culture and Competition: A Labratory Market Comparison between China and the West." Journal of Economic Behavior and Organization. Princeton, NJ: Princeton University Press.
Gintis, H. 2000. Game Theory Evolving: A Problem-centered Introduction to Modeling Strategic Interaction. Princeton NJ: Princeton University Press..
2The utility function for Homo equalis is
where x = (x1,...xn). Note that if there is perfect equality (i.,e xi = xj for all j) then the two negative terms in the utility function are zero.
Camerer, C. and R. Thaler 1995. "Ultimatums, Dictators, and Manners." Journal of Economic Perspectives 9, no. 2: 209-219.
Guth, W., and R. Tietz 1990. "Ultimatum Bargaining Behavior: A Survey and Comparison of Experimental Results." Journal of Economic Psychology 11: 417-449.
Blount, S. 1995. "When Social Outcomes Aren't Fair: The Effect of Causal Attributions and Prefernces." Organizational Behavior & Human Decision Processes 63, no. 2: 131-144.
3For example, randomly assigning people to teams. While such a division is random there appears to be affinity for people inside the group than to those outside the group.
Other posts that draw on this material are:
Are Austrian Economists About to Lose Their Cherished Position?
Robert F. Kennedy, Jr.: Prevaricator
More Problems with RFK, Jr.'s "Free Market" Environmentalism
Posted by Steve at February 27, 2005 03:34 PMNote that this does not make Homo economicus worthless or unimportant. In the public goods games for example a percentage of the players do play according to Homo economicus and this could be one reason why the cooperation in early stages deteriorates to the point where in later stages everybody is playing according to Homo economicus.
I think one point is that public goods depend on shared interest. Ie, most of the community has an interest in maintaining the public good. If there's no way to determine whether someone is overconsuming a public good, then it degenerates rapidly to pure self-interest. Even in the presence of some ability to police, the system can enter pure self-interest state.
But on the other hand, it can go the other way. After all, these systems came about somehow. So perhaps a real system would enter and leave different economic modes of operation where the collective economic behavior of the system can be completely different.
One thing that bothers me here is attempting to model the behavior of the player and the dynamics of the game using somewhat tautological assumptions. Ie, if the game assumes players are homo economicus, then the game has certain dynamics which I suspect support or give a competitive edge to behavior model over other behavior models. One doesn't cooperate in Monopoly (an economic game as well as a game), but that's because the rules of that game encourage that behavior.
Karl,
Well, actually that sort of thing may not matter that much. The deterioration of cooperation, some argue, is due to the cheaters and not cooperating is the only way to retaliate against them. However, allowing for things like cheap talk in these games and some method of retaliation then the cooperation becomes quite high and tends to stay there.
One thing that bothers me here is attempting to model the behavior of the player and the dynamics of the game using somewhat tautological assumptions. Ie, if the game assumes players are homo economicus, then the game has certain dynamics which I suspect support or give a competitive edge to behavior model over other behavior models.
No, I don't agree. Initially the experiments and experimenters were working off the assumption of Homo economicus and when the experiments returned different results it was a surprise. Most of these games have simple sub-game perfect equilibria, but they are rarely played. In other words, if you solved it via the standard game theory method and then let people actually play the game the two outcomes would different. Plus the results are pretty invariant. You see strange things even in one-shot games for instance, games with larger payoffs, different countries, etc.
Posted by: Steve on February 27, 2005 10:12 PMFrom July 12, 2004 post:
I think it would do Hoppe well to note precisely what these goods are as opposed to calling them something they are not.
Steve - thanks for providing so much information about Hans Hoppes. I didn't realize he was an intellectual force in addition to being this week's poster boy for "Whoever called economics the dismal science got it half right."
Is Hoppe an anarcho-capitalist? It's a weird term, most anarchists I know despise capitalism. Does anarcho-capitalism mean fundamentalist capitalism? Complete and total blind faith in markets?
Could you summarize the austrian asshole's views in a paragraph or two so I could better understand where the prick's coming from?
Thomas Carlyle, but not in reference to Malthusian population dynamics as has become canon over the years, but rather that he preferred slavery to free labor markets. Relevant passage from the above link:
Carlyle puts the view that 'work' is morally good and that if a "Black man" will not voluntarily work for the wages then prevailing he should be compelled to work. He writes of those who argued that the forces of supply and demand rather than physical coercion should regulate the labour market that: "the Social Science ... which finds the secret of this Universe in supply and demand and reduces the duty of human governors to that of letting men alone ... is a dreary, desolate, and indeed quite abject and distressing one; what we might call ... the dismal science" (Volume 11, p 177). He also uses the term "dismal science" in a derogatory way a number of times later in the work, where it is lumped together with other unwelcome (to Carlyle) features of the political scene as "ballot boxes", "universal suffrage" and "Exeter-Hall Philanthropy". At one point he tells us that it is unwise to have a situation where "supply and demand [is] the all-sufficient substitute for command and obedience among two-legged animals of the unfeathered class" (p 186).
Posted by: Timothy on February 28, 2005 08:55 AM
More from the same source:
In short, Carlyle was of the view that compulsion, rather than market forces should regulate the supply of labour on plantations in the West Indies because the laws of supply and demand are not appropriately applied to the relationship between White and Black as they are contrary to "their mutual duties" (white = master and black = servant) as ordained by "the Maker of them both" (p 207). In Carlyle's opinion: "declaring that Negro and White are unrelated, loose from one another, on a footing of perfect equality, and subject to no law but that of supply and demand according to the Dismal Science", "is clearly no solution" to the problem (ibid). Instead, Carlyle offers life-long servitude "after the manner of the old European serfs" as the best solution because in such a regime, "it ought to be rendered possible, for White men to live alongside Black men, and in some just manner to command Black men, and produce West Indian fruitfulness by means of them" (ibid).Dismal because supply and demand should determine labor supply, not whips and coercion as Carlyle would've had it. Personally, I think all Economists should feel proud to be called dismal by such a man. Posted by: Timothy on February 28, 2005 09:01 AM
Steve and Karl:
The point about how long the game is played is one of the crucial parts that, I suspect, is less well-understood at times.
As Axelrod noted in his book on game theory (iirc), iterative games have different dynamics than those which are single-shot. Indeed, part of Axelrod and Schelling's theories noted that, if you know the game is going to end, even if it is still somewhat in the future, you wind up seeking greater gain in the short term. Only by not knowing if the game is going to end is the iterative aspect fully realized.
Posted by: Dean on February 28, 2005 11:16 AMDean,
Again, while what you note is true, the level of cooperation even in one-shot games is not in line with Homo economicus. In fact, IIRC, there was some research where the groups were randomly mixed so that retaliating had little hope of improving future play for the retaliator, but retaliation still took place even at a cost to the retaliator. That doesn't sit well with standard theory.
Posted by: Steve on February 28, 2005 11:39 AMbirdbrain: Steve Verdon has a very nice post today on the various ways in which experimental economics (a.k.a. behavioral economics) is changing the way economists think about behavior.
Yikes - the birdbrain doesn't even understand the difference between John Nash, Vernon Smith and experimental economics and Savage, Kahneman, Tversky and behavioral economics.
What a dimwit.
I tried to read your post but I couldn't detect a point. Sorry.
It is pretty much the point of some of your JDM posts Deb. Just to give the reader some insight to something they may not have seen in a 101 type class. Most people who have suffered through an econ course have heard of economic man, Homo economicus, but they haven't heard of these new developments.
In fact, I was just reading a post of yours about loss aversion and how you concluded that just because humans don't always behave according to Homo economicus doesn't mean they are irrational. Was that the point of your post? If so, maybe I should use your retort: Your meme to word ratio is simply amazing. You can write so much and say so little. The truth is I liked the post. For once you weren't going crazy on economists calling the entire profession a bunch of morons. But for me the answer was obvious, but somehow...I managed to resist the urge to post a snide comment.
Posted by: Steve on February 28, 2005 04:12 PMIn fact, I was just reading a post of yours about loss aversion and how you concluded that just because humans don't always behave according to Homo economicus doesn't mean they are irrational.
That is the point, Steve. Allais was the first JDMer, although he's not described that way. He was ADAMANT that his counterexample to the axioms of EU did not only challenge its descriptive validity but also its normative validity.
But economists worship the axioms of EU the way Christians and Jews worship an alleged guy in the sky. Kahneman and Tversky didn't have the nerve or the neurons to go as far as Allais and debunk EU completely (descriptively AND normatively).
So if you agree that loss aversion and other violations of the axioms of expected utility theory a la von Neumann, Morgenstern and Savage can be RATIONAL, then we are indeed on the same (lonely) page.
Posted by: lefty on March 1, 2005 06:07 AMit is quite often the case that people don't make the right choice given that the model of human behavior is expected utility maximization. Of course this does not mean that Homo economicus is everywhere and anywhere a bad model for human behavior.
Well, actually, yes it does. And the point I often make is that it is not necessarily a bad thing that people don't think the way economists say we ought to.
And Muslims, lefty. You forgot that Muslims worship the same "guy in the sky."
Or is it too un-PC to note that?
Posted by: Dean on March 1, 2005 07:36 AMBut economists worship the axioms of EU the way Christians and Jews worship an alleged guy in the sky.
I disagree. Economists are quite aware of Allais and the problems. Some are looking at new ways of looking at choice under uncertainty.
Kahneman and Tversky didn't have the nerve or the neurons to go as far as Allais and debunk EU completely (descriptively AND normatively).
In your opinion.
So if you agree that loss aversion and other violations of the axioms of expected utility theory a la von Neumann, Morgenstern and Savage can be RATIONAL, then we are indeed on the same (lonely) page.
I don't think we are both on the same lonely page, but perhaps on the same page in terms of choice under uncertainty. The loneliness is probably due to other factors.
Well, actually, yes it does. And the point I often make is that it is not necessarily a bad thing that people don't think the way economists say we ought to.
No it doesn't, because in certain instances people do behave according to Homo economicus. Even in instances where there are things like uncertainty or other issues such as public goods some people play according to standard economic models. Does it disprove economics? In a trivial sense, but it means that any improvements will have to account for both what the economists have posited and what other researchers such as Kahneman are finding.
It is like the evolution/creationism debate. You are like the creationist. You seem think that simply because you can prove that not everybody behaves according to Homo economicus that all of economics is utterly worthless. Creationist figure if they can punch one hole in evolutionary theory then all of evolutionary theory is false and Creationism is true be default. This is false. What it means is that the new theory that addresses the short coming of current evolutionary theory will have to also explain everything current evolutionary theory already explains.
Similarly any new model of human behavior will have to account for the results from standard economic theory as well as the results from behavioral economics. It doesn't mean throw out the baby with the bath water. If you indeed insist on the later no wonder you are in a lonely place research wise.
Posted by: Steve on March 1, 2005 09:55 AMSteve: You are like the creationist. You seem think that simply because you can prove that not everybody behaves according to Homo economicus that all of economics is utterly worthless.
Not quite, although it's an interesting analogy. I don't think economics is worthless - economics makes prescriptive claims about how society should be designed - which things should be market based and which things should not, how markets should be designed and regulated, etc. There's something very right and very powerful about Adam Smith's invisible hand.
My gripe is that economics is systematically biased - that it would be enormously improved by taking Allais, Kahneman, etc. SERIOUSLY. Right now, it's a quaint hobby that economists dabble in. The VAST majority of economists are entirely faith-based (assume rational choice theory) instead of being evidence-based.
Homo economicus is a heuristic - an approximation of the human mind. It's not a completely false approximation. It's not useless. But the world cannot survive much longer with the myth that people and markets are rational until proven otherwise.
Posted by: lefty on March 1, 2005 10:16 AMHomo economicus is a heuristic - an approximation of the human mind. It's not a completely false approximation. It's not useless. But the world cannot survive much longer with the myth that people and markets are rational until proven otherwise.
I think you are making a bit of a mistake here. The markets might be quite rational, they are just irrational given the assumptions many economists make. For example, Homo reciprocans looks irrational when viewed through the lens of Homo economicus, but that by itself doesn't make Homo reciprocans irrational. A point which you have made. By the way, I like the poison oak story you have on this, a very nice way of making this point.
Other than that I agree that there can be problems in terms of policy by not having the right "model" for how people work. And it isn't just the Righties that can cause problems as I noted here. The envorimentalists seem to be all to ready to implicitly work with the Homo economicus view of humanity. This can lead to bad policy that is actually bad for the environment.
Posted by: Steve on March 1, 2005 10:40 AMNo, I don't agree. Initially the experiments and experimenters were working off the assumption of Homo economicus and when the experiments returned different results it was a surprise. Most of these games have simple sub-game perfect equilibria, but they are rarely played. In other words, if you solved it via the standard game theory method and then let people actually play the game the two outcomes would different. Plus the results are pretty invariant. You see strange things even in one-shot games for instance, games with larger payoffs, different countries, etc.
Good points, but I didn't intend to say what you apparently thought I said. My point isn't that humans play or don't play optimal solutions, but rather that a number of models contain assumptions about what strategies the players employ. I'm thinking, for example, about Black-Scholes and related models that rely on the absence of arbitrage.
Incidentally, what games are humans typically near optimal at? I wonder also if connotation can affect game play. Ie, would players play better poker with the same eventual payoff, if the chips were pretty, shiny metal rather than little spooky plastic skulls?
Posted by: Karl Hallowell on March 2, 2005 06:49 AMUh, Karl, just what kind of chips do you play poker with?!?
Posted by: Dean on March 2, 2005 07:21 AMMy point isn't that humans play or don't play optimal solutions, but rather that a number of models contain assumptions about what strategies the players employ.
Yes, and this is what this research is looking at. The Homo econmicus model basically holds that people should maximize their payoffs. While some people do, not all people do. So, a new model is needed since people deviate from the predicted behavior of the model.
Incidentally, what games are humans typically near optimal at?
When the proposer is a computer and this is known to the (human) respondent. In this case, the rejection rate falls considerably.
Posted by: Steve on March 2, 2005 09:33 AMSteve:
Is it that people do not maximize their payoff, or that people maximize different payoffs?
Frex, I can imagine that for stockholders A and B, A chooses to maximize the amount of money he makes, but B chooses to maximize the amount of "good" that the company does (be it measured in terms of jobs preserved, environment not dispoiled, etc.).
Both A and B are maximizing returns, but they are doing so on different axes (or in different terms).
Does homo economicus assume that people maximize only for money---insofar as that is the only thing that A looks at, that might be true, but there will be B and C and D for whom it is not?
Whereas if homo economicus assumes only that people will maximize according to what they value (but that what they value might be something else---karma, money, free time), i.e., that people engage in cost-benefit calculations in order to maximize something that they value, that would still seem to be true for all people. Rare is the person who would seek to minimize that which they actually value. Even the altruist presumably derives some kind of benefit (perhaps purely spiritual or existential) from their altruism?
Posted by: Dean on March 2, 2005 01:06 PMExcellent question, Dean. Take the ultimatum game. Player A has $10 to split between A and B. If B accepts A's offer, A and B leave with money. If B rejects A's offer, A and B leave with bubkas.
A simple game theory analysis suggests that A should offer a penny and B should accept it. But most A's offer way more than a penny and it's a good thing since most B's reject offers less than $3.00.
Imagine you are B. A offers a dollar. You can accept the offer and leave with a dollar and let A leave with nine. Or you can say BUCK YOU to A and both leave with nothing. There's a utility to saying BUCK YOU to a selfish jerk.
The problem for game theorists is this:
If you use the narrow definition of utility maximization where you assume all that matters is monetary payoff, the theory is blatantly false AND normatively unjustified.
If you use the broad definition of utility where people have utility for altruism, saying buck you, etc., the theory is normatively justified but unfalsifiable.
Posted by: lefty on March 2, 2005 01:55 PMIs it that people do not maximize their payoff, or that people maximize different payoffs?
Yes. Note that the second part implies the first part. That is if I'm also interested in fairness as well as being kind (i.e. I want to be kind to my fellow man, but I also don't want to be taken advantage of) I might have a different objective function than simply maximizing my own monetary position. Hence I am not maximizing the payoff and I am also maximizing some different objective.
Does homo economicus assume that people maximize only for money
No, more accurately Homo economicus is purely selfish in a very narrow sense of the word. If a person acted totally in accord with Homo economicus most people would consider that person a sociopath. So as Deb says it is a heuristic, a model that is a simplified version of reality. Some people do seem to base some of their behavior on this model. They are the early defectors in things like the public goods games, the common resource pool games, etc. These are the people who will periodically defect in prisoner's dilemma to see if perhaps you are too trusting.
So it is important to have a model for this, to do research on it, but also one has to remember that it isn't the only model.
If you use the broad definition of utility where people have utility for altruism, saying buck you, etc., the theory is normatively justified but unfalsifiable.
One does have to be careful with expanding the utility function so that it becomes all encompassing. I think this is where evolutionary game theory can come in handy. Homo reciprocans does suggest that at later stages in a game if a player is defecting routinely and early on, that defecting will become the norm even for Homo reciprocans as Homo reciprocans will seek to punish the early and frequent defectors. This is what is observed in many games such as the public goods game. Initially there will be a fairly high level of cooperation and then later that cooperation decays till defect is used by all players. Toss in cheap talk (i.e. non-binding pre-play talk) and a retaliation mechanism (that has a cost) and cooperation is much higher and, IIRC, can be sustained for longer periods.
Posted by: Steve on March 2, 2005 04:21 PM