Value Competition

Final Jeopardy! and Supply Chain, Continued—Continuous Choice between “Confident” and "Timid" Proposals

To make the supplier-buyer scenario described in the last post directly analogous to the Final Jeopardy! betting situation, one needs to convert the dichotomous choice between confident and timid proposals into a continuous choice that parallels the players' decisions to bet anything from nothing to all their money.

Additional specifications for the continuous variable case:

1) The advantage of supplier A is equal to x.  X is a quantity known to both suppliers.  (This condition parallels the known lead that player A has over player B going into the Final Jeopardy! round.)

2) A will win unless B's proposal exceeds A's in value by at least x. (This condition parallels the need of the trailing player in Final Jeopardy! to make up the difference with the leading player.)

3) A and B can both submit proposals on a range from 100% timid to 100% confident.   A 100% timid proposal takes no risk and has a baseline value of 0; a 100% confident proposal takes maximum risk, and has value y if successful and -y if unsuccessful.  All other proposals have an intermediate value; for example, a proposal that is 75% confident has a value of .75y if successful and -.75y if unsuccessful.  (This parallels the ability of both players to bet any or all of their money.) 

4) Y is greater than x.  That is, there is no guaranteed win for A.  (This is equivalent to a Final Jeopardy! round in which the second place player has more than 50% of the leader's total.) 

The continuous variable case is not as straightforward to analyze in rational choice terms as the dichotomous variable case is.  The mixed Nash equilibria are parallel to those described earlier for the dichotomous variable case: The leading player should tilt toward confidence, while the trailing player should tilt toward timidity.

 

 

Final Jeopardy! Betting as a Competition between Two Suppliers—Dichotomous Choice between “Confident” and "Timid" Proposals

Suppose there are two suppliers A and B who are competing to win a contract from a buyer for consulting services.   Supplier A is favored by the buyer (that favoritism may be because of higher quality, a prior relationship, or other factors).  If the suppliers’ proposals are of equal value, the buyer will choose supplier A.   Otherwise, the buyer will chose the supplier with the higher value proposal.   The suppliers compete under the following conditions:

1) The suppliers may submit either a “confident” consulting  proposal for improving the buyer’s operations that covers a wide range of areas or a “timid” proposal that presents more detail on a single area;   

2)  A proposal will be viewed as either  successful or unsuccessful by the buyer;

3)  Confident and timid proposals are equally likely to be viewed as successful;

4)  The probability that a proposal will be viewed as successful is equal for the suppliers and is greater than .5;

5)   There is a positive correlation between the probability that A’s proposal will be viewed as successful and the probability that B’s will be viewed as successful (this condition may be understood in terms of the state of the buyer’s operations being either one in which it is easy to make improvements  or one in which it is difficult to make improvements, with the state being private information that is known to the buyer but not to the suppliers);

6)  The ordinal value of proposals to the buyer is as follows, from highest to lowest:  Successful confident, successful timid, unsuccessful timid, unsuccessful confident.

From the above conditions, one can conclude that the nonfavored supplier B can win in three ways: By submitting a successful confident proposal while A submits a timid proposal or an unsuccessful confident proposal, by submitting a successful timid proposal while A submits a confident or timid proposal that is unsuccessful, or by submitting an unsuccessful timid proposal while A submits an unsuccessful ambitious proposal.

Now, the rational choice analysis: Assuming A and B are both acting to maximize the probability of winning the contract, both have an incentive to play a mixed Nash strategy that cannot be exploited by the other supplier.   A numerical example: Assume the overall probability of a successful proposal is .75 and that the conditional probability the other supplier’s proposal will be successful if one’s proposal is successful is .8.  In that case, mixed Nash for the favored supplier A is to make a Confident proposal C with p = 6/7 = .857 and a Timid proposal T with probability p = 1/7 = 143.  Mixed Nash for the nonfavored supplier B is to make a C proposal with p = 1/7 = .143 and a T proposal with probability p = 6/7 = .857.

The intuition behind the high level of timid proposals from the nonfavored supplier B in the mixed Nash equilibrium for the two players can be described as follows: Intuitively, the favored supplier A should mostly submit confident proposals, which are likely to be successful and to guarantee the contract for A.  Knowing that A will mostly be confident, B should mostly be timid.  If A submits an unsuccessful confident proposal, B will win the contract for sure with a timid proposal.

Reflections on an Unread [Now Semi-Read] Article

Read Jonah Lehrer's piece in the New Yorker on the kerfuffle between E.O. Wilson and his co-authors in a recent math-laden article in Nature (unread by me) and the defenders of Hamilton's inclusive fitness model of altruism  formerly espoused by Wilson himself...was profoundly impressed by Wilson's readiness as an 82-year old to reexamine and revise his assumptions...

I'm strongly with Wilson in wanting an alternative to approaches such as Hamilton's that focus too narrowly on kinship (and also to approaches such as Trivers' that focus too narrowly on reciprocity).   I believe that altruism in the form of taking into account at least some of the payoffs of others as well as one's own payoffs is a reality for many humans and nonhumans in situations involving strangers and one-shot games.  Given that prior belief, I think there is real value to an approach to modeling altruism that explains how at least some kinds of it can be an evolutionarily stable strategy in situations involving non-relatives and one-shot interactions.

But allowing that I haven't read the Wilson, Nowak, Tamita article, I'm quite doubtful that their high-tech approach is the way forward for biologists.  I think the way forward will instead involve a low-techThomas Schelling-Robert Frank inspired understanding of the ways in which selective altruism can be helpful to the altruist in leadership and trust games that are as or more important in the real world than prisoner's dilemmas in which altruism, selective or otherwise, is harmful or at best unhelpful.

....

[Decided I should at least skim the Nature article...checking it out...]

I skimmed the article...I'm still doubtful, not because of the story (defensible nests/strong groupishness/altruism), which seems good (not that I would know either way), but because the story seems to be cast in prisoner's dilemma terms...if so, the mainstream defenders of inclusive fitness/reciprocity, etc. can always make the standard rejoinder that as well as altruists may do with other altruists they will not do well with egoists absent the ability to identify and punish them...the way to get out of this box I think is to tell a story that decenters the Prisoner's Dilemma in favor of coordination/leadership/assurance games in which certain types of altruists take egoists to the cleaners in a one-shot game rather than the other way around...

 

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A Self-Correction on Game Names

Since 2010, I’ve classified symmetrical 2 x 2 games into four categories: Harmony games, Trust games, Cooperation games, and Leadership games.   That terminology is reflected in any number of posts on this blog, as well in other matrix-based work I’ve done.

I’m still very good with Leadership, but now have a quibble about Trust and bigger problems with Cooperation and Harmony.

The idea that hit me earlier today and that led to my sense I’ve got some of the names wrong: Games can/should be labeled on the basis of the attribute on the part of one or both of the players that is efficacious in optimizing in the game.

Applying the idea to the four basic types of games:

Leadership is a good term for the class of games such as Battle of the Sexes in which the best or equal best outcome is achieved through leadership by one player that is respected b y the other player.

Trust is an all right but less than best term for a class of games such as Stag Hunt in which the best outcome is achieved though one player being able to assure the other that he/she will play the jointly maximizing strategy that is best for both players.   Trust can be reasonably taken to imply a reciprocity that is not necessary for the optimal outcome in these games, in which only one player need assure the other of his or her intentions.   Assurance gets at that unilateral concept better than trust, and is hence a somewhat better label  for these games.

Cooperation with its highly generic, imprecise associations now strikes me as a weak and overly general term to apply to a class of games consisting of the Prisoner’s Dilemma and one version of Chicken.  In the PD and the relevant type of Chicken, the distinctive game property is that altruism on the part of both players solves the game cleanly.  Absent mutual altruism, these games are very tough ones.  The term for these games that now seems right to me is thus Altruism, or—to be more precise if also clunky about it—Mutual Altruism. 

Finally, Harmony seems like a weak way to describe the games in which the dominant strategy for egoists results in the jointly best outcome.   Simply put, the attribute that works to solve these games is Egoism.  (Note: Altruism fails to work well in a number of them, though it does solve a number.)   Hence, I would call these games Egoism games.

So, the new tetrarchy of labels for games: Egoism, Altruism, Assurance, and Leadership.

We’ll see how long this typology lasts :)*

*Maybe not too long...it occurs to me that using Egoism and Altruism as names for games creates an issue.  In a few easy games both altruism and egoism solve the game; calling all these games egoism games involves an arguable bias in favor of that attribute.  We could call those games Rationality games to indicate that either egoistic or altruistic calculation works to solves these games.  In that case, the tetrarchy becomes a pentarchy, with Rationality added to the other four qualities that solve games.  That seems right, foir now :)

Mixed Nash continued

Following up on the four-person game analyzed in the previous post...

Memo to self: If the Pascal-Bentham-Whitman-Nash type that optimizes in two-player games and holds its own against egoists is a truly groovy type, it should also optimize and be an evolutionarily stable type in the four-player game in which you play either 2 or 10 and get either $2 or $10 except that if all four choose 10 no one gets anything.

As a two-person game, this is a leadership game, and I'm morally sure (that is, I think I know but don't really know) that P-B-W-N will optimize because the game is similar enough to ones I have analyzed.

As a four-player game...well, I haven't really thought about multi-player games for P-B-W-N purposes, basically because the tech gets out of my league...

A note or two...the .928 probability of playing 10 that is mixed Nash for four egoists is decidedly suboptimal...a little SAT-style plug-in suggests that the optimal level of playing 10 is somewhere between .5 and .75...so P-B-W-N if it is an optimizing type in the 4-player game should play 10 at that lower level...to meet the criterion of being an evolutionarily stable strategy, it should also be resistant to egoism...

I'm genuinely interested (i.e., no moral certainty here) as to whether P-B-W-N is optimizing and an ESS in the 4-player game...a to-do for when some time opens up after getting my SBE paper for 3/1 done...

 

 

How much do you believe in mixed Nash?

Hat tip to my son for telling me about the game below...

Suppose there are four players who can independently choose either 2 or 10 and will receive $2 for 2 and $10 for 10 EXCEPT that if all four choose 10 they will all get zero.  

In that game, three players choosing 10 and one choosing 2 is Nash, in that no one has an incentive to deviate knowing the other's choices.  Working out mixed Nash is trickier...the intuition is that if the others play a strategy mix that leaves you with a payoff of x from playing 10 and the same payoff of x from playing 2, that mix is Nash...so let's see...your payoffs are equal if there is a .8 chance the others will all choose 10 and a .2 chance they won't...the tech is then that the others "should" play 10 with a p = p to the third power = .8 or about .928 if mixed Nash is treated as normative.

Now, I personally believe in mixed Nash as a powerful guide to unlocking social reality.  (Maybe that belief is quite wrong--but it makes sense given the academic project I'm engaged in, which tries to marry mixed Nash with social preferences like altruism and competitiveness.) 

Give a belief in mixed Nash and egoistic preferences, one would expect that a lab experiment would show people playing 10 a bit over 90% of the time.   My intuition is that that would in fact be borne out. 

(A qualification to reflect my strong belief in analyses that consider social preferences: The prediction above is dependent on the framing of the experiment implying that self-interested preferences--as opposed to social preferences of one kind or another--are the relevant ones in the situation.)

 

Altruistic Values by High and Low Productivity Employees

Under Bob Frank's model of relative position, more productive employees pay for their higher position relative to their less productive peers.  The result is that pay within a group of employees, such as faculty in an academic department, is substantially more compressed in an egalitarian direction than their productivity.  The microfoundational basis for why higher productivity employees pay is that they value deference and higher rank, which they cannot get as effectively by banding together with other high-productivity employees as they can by working with peers who are less productive.  Although Frank's model can be understood in terms of pure egoism, it offers a much more powerful lens into reality if one accepts the assumption--a highly plausible one, in my view--that people have competitive preferences as well as egoistic ones. 

The interaction between the high productivity employees and their low productivity peers can be modeled as a leadership game.  We may assume that both players do badly in the event of a leadership conflict in which they cannot agree on whether, or how much, the high productivity player pays for rank; the leader does best and the follower does adequately if they agree on an arrangement favoring either the high or the low productivity player; neither side does too well or too badly if neither side asserts itself.  

It is intuitive that if either side can commit itself to leading, it has an advantage over the other.  On the face of it, the reverse is true for a player who embraces the idea of following the other player's leadership.  Such a player--a high productivity player who likes the idea of paying a high tax for his or her rank or a low productivity player who believes that the high productivity player should not have to pay for a leading position--would seem to be at a major disadvantage in the leadership game.

The altruist who values following the other's leadership is indeed at a disadvantage if his or her preferences cause her to play following as a dominant strategy.  But it is not likely that valuing the other's leadership will in fact produce that counterproductive result.  If the other player follows, the altruist may be assume to have a preference for leading, just as the egoist does. W

What valuing following the other's leadership is very likely to do is to create a game in which the other player realizes that the altruist has a very large difference between his or her payoffs when the other leads and a smaller one when the other player follows.  The counterintuitive logic of mixed Nash equilibrium leads us to expect that the other player will therefore follow most of the time, resulting in the altruist receiving substantially better payoffs than the other player. 

Hooking back to Bob Frank and his model: Bob himself is a highly productive academic who seems very much fine with a high tax.  Some of us--no names mentioned!--aren't as productive and have ethical qualms about a tax paid by the highly productive.  The suggestion here is both the altruistic highly productive employee--RF we'll call him or her--and the altruistic less productive employee--WE we'll call him or her--may have a belief system that plays out to their advantage.  The altruistic and highly productive RF may find himself getting a bigger raise from the faculty peer evaluation committee than if he were an egoist; the altruistic and not so productive WE likewise may do better in exacting a tax from RF that goes to himself than he would if he were egoistic.

Three Games in Which Altruism Can Be Advantageous

[more adv. alt. notes]

In the Prisoner's Dilemma, the egoist has a dominant strategy of defecting.  Given that, there is no way that altruism in the Dilemma can help the altruist in a one-shot game.  The egoist will defect regardless, and the altruist will wind up with either his or her worst or second possible payoff.  (footnote--There are five additional games in which the egoist also has a dominant strategy and thus cannot be affected by altruism on the part of the other player.  In those games, though, unlike the Prisoner's Dilemma, the dominant strategy for the egoist is either clearly optimal for both players or very possibly optimal.  Accordingly, these five games--which could be termed Harmony Games because of their benevolent payoff structure--do not present  the significant difficulties for egoism that the Prisoner's Dilemma does.)

In six other games, though, the egoist has no dominant strategy.  In all of these games, altruism can be potentially useful to the altruist as well as to the two players together.   Three of the six games may be called assurance games.  In these games--Trust, Schelling, and Stag Hunt are the labels that will be used here--both players earn their highest payoffs if they both trust.  But if the other player does not trust, you are better off if you also do not trust.  (footnote--The three games differ among themselves in two major respects.  In two of them--Trust and Stag Hunt--if you trust when the other player does not you get your worst possible payoff, while in the third, Schelling, you get your third worst payoff.  In two of them--Schelling and Stag Hunt--both players get their second-best payoffs from both not trusting and going it alone, while in one, Trust, both players get their third-best payoffs from both going it alone.)

In the assurance games, altruism has the potential to be helpful if it leads to the altruist committing to trust.  Such a commitment leads the egoist to trust, since trust-trust is the best outcome for both players. 

Interestingly, full altruism does not do the trick in leading to commitment to trust.  As we will explore further, only a form of selective altruism that values the trust-distrust payoff for the trusting player highly will work.

The Combined Value of the Idealistic and Practical Types

[more advantageous altruism notes]

One with a Jesus/Kant idealistic bent will likely find the "we're all right" upbeat practicality of Smith's take on human nature complacent and jejune at best, and at worst complicit in a crass culture of materialism and moral squalor.  On the other side of the moral fence, one with a Smith/Hume bent will likely feel a combination of respect and amused disdain for the soaring aspirations of Jesus and his secular idealistic successors.  This paper, though it is written from within the Smith/Hume tradition, assumes that both the anguished, self-critical sprit of Jesus and Kant and the optimistic, practical spirit of Smith and his great friend and contemporary David Hume are key constituents of positive human nature and of normative ethics.

Making a broad case for the viability of the assumption stated in the last paragraph would be a very large project indeed.  The point to be made here is a smaller, subsidiary one: The practical, limited altruism of the Smithian type can be usefully challenged at the margins to be nicer and more forgiving by the idealistic altruism of the pure Jesus type. The practical Smithian will shrug off the pure Jesus type's idealism when doing so is costly, but will be very open to changes in a more altruistic direction when they can be incorporated into a practical routine of reciprocated trust and generosity that is helpful to both players and to society.  The pure Jesus type, as opposed to the partially forgiving Jesus type of this paper, lacks the skills to differentiate between advantageous and disadvantageous altruism that the Smithian type has in abundance.  But the strong Jesus type usefully challenges the Smithian type to be altruistic in new ways that benefit the Smithian player, the egoist, and society.

A parallel should be noted here between the practical and idealistic sides of Greek virtue ethics represented by Aristotle and Plato respectively and the practical and idealistic sides of modern ethics represented by Smith/Hume and Jesus/Kant respectively.  The all too practical spirit of Aristotle's great-souled man was usefully challenged by the idealistic spirit of Plato's guardian.  To be sure, both were elite types who dominated the ordinary person; to adopt Nietzsche's provocative terminology, Greek ethics was master morality, not slave morality.  But if one can detach oneself from both disdain for the elitism of the great Greeks and from defensive upholding of their eternal wisdom, one can see in the combination of Plato's and Aristotle's versions of advantageous altruism a desirable check and balance to leaders who are either unrealistic fools committed to ideals with no purchase on reality (for Aristotle whatever else he is is no fool, and no one to suffer fools) or all-too-practical seekers of advantage with no ideals (for Plato whatever else he is is a true believer in ideals). 

With the Greeks viewed with a proper measure of sympathy and dispassion, one may also gain perspective on our current situation.  For all the aspirations of modern ethics of both the Smith/Hume wing and the Jesus/Kant wing to be a universal morality rather than a master morality of those who lead and rule--aspirations which here will be assumed to be entirely correct, and which I share--modern ethics is in part a guide to leaders and rulers, just as Aristotle, Plato, Confucius, and Lao Tzu were. 

The elevated position of the leader now as in classical times is justified, if it can be, by a greater concern on the part of the leader for followers, for others in general, and for the whole than is found among more egoistic, narrower, less elevated non-leaders.   Again, if one can detach oneself both from indignation at the degree to which modern ethics remains elitist and of indignant defense of elitism as an eternal verity, one can see in the combination of Jesus and Smith a highly useful set of restraints and incentives for modern leaders that is lacking in either type alone.  Smith will not be a fool. much as he is far nicer than Aristotle about suffering them. Jesus will not suffer hypocrites, much as he is far nicer than Plato about those who fail to live up to ideals.  In the combination of the two modern types as in the combination of the two classical types, one finds usefully ambivalent leaders who are neither slaves to ideals nor to advantage.

Symmetrical and Asymmetrical Sympathy

If people are sympathetic as a matter of character, it would not seem that symmetry in a game should matter, unless it is the case that symmetry in fact makes sympathy more advantageous socially and/or individually.  On the other hand, if people are situationally sympathetic, symmetry is likely to matter.  Being sympathetic to the other's feelings of fulfillment for coordinating properly on a highest joint value outcome and regret for failing to do so is facilitated when the other has a situation parallel to one's own.  When the situations are not parallel, on the other hand, there may well be logical ways to coordinate other than the highest joint value outcome.

Consider the cardinal payoffs matrix below, in which Row has Battle of the Sexes payoffs and Column has Leadership payoffs:

                     Follow        Lead

    Follow         1, 2           3, 5

    Lead           4, 3           2, 1

The highest joint value outcome is for Row to follow and Column to lead.   On the other hand, there is a certain focal point logic stemming from the asymmetry of the game that favors Row leading--in which case Row is guaranteed avoiding his/her worst payoff--and Column following--which guarantees that Column will avoid his/her worst payoff.  By comparison, coordinating on the highest joint value outcome should be easier if both players have the same BOTS or Leadership payoffs, with the only asymmetry being that Row following and Column leading has higher value than the reverse.  (The completely symmetrical standard formulations of BOTS and Leadership are tricky for another reason: Absent an HJV outcome, there is no good reason to coordinate on the upper right with Column leading as opposed to the lower left with Row leading.)