Thursday, October 16, 2014

Inspiring Words

Leadership is, to a great extent, the gentle art of persuasion. Leaders inspire others to follow them, to work for them, sometimes even to give up their own lives for them. How do they do it? Partially by example to be sure, but even here persuasion has a role to play. When we say that Jeff Bezos lives the leadership principles articulated and promulgated at Amazon, it makes the valid point that individuals credit others for how they behave, but conveniently ignores the fact that it was Bezos who articulated and promulgated the principles in the first place.

One of the most striking examples of leadership purely by the powers of persuasion was the rise of Barack Obama. Obama, for all his subsequent faults, was matchless in using words to inspire many thousands of young people who had never even voted previously to give him money, work for his organization, and persuade others to vote for him. Even more remarkably, he duplicated the feat again four years later, by most accounts, after spending much of that period by not leading by example. This no doubt overstates the power of his oratory for he had a remarkably savvy organization helping him to vacuum up all that money and effort, but others less gifted have had equally efficient teams, yet achieved nothing like Obama's success.

What can game theory tell us about the power of leaders to persuade? The answer, if we are to be entirely honest, is surprisingly little. A large part of the problem is that communication in the world of game theory is almost entirely informational, but persuasion, while it will certainly draw upon and convey some information, taps into something much deeper and less purely transactional than what one might learn from hearing the local weather forecast. This is not to say that information-centric communication, or persuasion, is uninteresting, rather that it somewhat misses the boat if we truly wish to understand why some individuals are hailed as visionaries while others, offering the same facts and conclusions, are not.

To get the flavor for the bloodless world of communications viewed through the lens of game theory, consider the following problem. A political leader wants to convince a supporter/follower to perform a certain action. The right action depends on something called a state variable, which you might think of as a shorthand description for a set of factors, political, economic, cultural, etc., that influence what a reasonable person would conclude is the correct action. To keep things simple, suppose that one state, which we will call low, represents a low political threat environment. Some action is needed to secure victory, but not too much. The other state, which we will call high threat, requires frenzied activity such as massive calling campaigns to get out the vote, and so on.

The follower wants to do the "right thing" for the leader, but knows little about political threats, and so on.
Knowing nothing about the state, our follower will elect some intermediate range of activity, imperfect for either state but somewhat helpful in both.

Now for the rub or, as we in the profession write, the "central tension of the model." The leader too wants the follower to do the right thing, but prefers that she do more political activity in either state. The degree of difference in views about how much activity to perform in each state represents the conflict between the two. Our leader's job, then, is to inspire his followers to do more than they otherwise would, but this will prove difficult since, in game theory land, the only trump card the leader holds is his knowledge of the state.

So, to inspire his supporters, our leader comes to town and makes a speech attempting to rally them to, in the leader's eyes, the right amount of activity. How does this speech go? What should our leader say? The answer, it turns out, depends on several factors, none of which feel (to me at least) very much like leadership.

Scenario #1: Free Speech
Suppose that our leader is free to say whatever he likes. He can lie about the state, exaggerating the political threat when it is, in fact, low or do the reverse, reassuring followers that there is little to worry about. Or something in between, saying that he's not sure. Or our leader can stonewall, give his standard stump speech, shake hands and kiss babies, Purell his hands and lips afterward, and go home.

So what does he do? To answer this question, we need to make certain assumptions about what, exactly, the followers know. Suppose they know that the leader indeed knows the state and, importantly, they also know that the leader wants them to do more of the activity in each state than they themselves prefer. In the happy scenario, the leader only wants the followers to do a little more in each state, so he informs them about the state truthfully and then harangues them "exceed themselves" or to "go beyond" or something like that. He gets his applause and leaves, satisfied at a good night's work.

Game theory, however, offers the exceptionally dreary conclusion that, no matter how powerful the words of inspiration, no matter that our leader is a Shakespeare or a Churchill, the followers do precisely what they had initially planned to do in each state. They are grateful for the information, but they can hardly be said to be inspired. Ironically, this situation is, in fact, the best our leader can hope for.

Let's rerun the speech but now imagine that the leader's vaunting ambitions create a vast gulf between his preferred activity level in each state and their own. So our leader steps up to the microphone to the hushed crowd and proceeds to speak of crisis--the threat level is high, the stakes are huge, and it's all up to you, the supporters to make the difference. This address, Shakespearean in its majestic, soaring phrases, send chills down the spines of the audience. The crowd roars. They will do it. They will rise to the challenge. They will be the difference-makers. No activity is too much. Our leader, drenched in sweat from the effort, steps down from the lectern and is congratulated for his remarkably moving address. The lights in the auditorium go down, and everyone goes home.

When his supporters get up the next morning, they do...exactly what they would have done if the leader had never shown up in the first place. In game theory land, people are cynics. While the audience may have been moved in the moment, on reflection they realize that the leader makes this same speech everywhere, to all his followers, whether the state is high or low. The talk, for all its pageantry, rings hollow--full of sound and fury, but signifying nothing. Why such an uncharitable view of the leader? The answer is that his own aspirations get in the way. Since he wants a high level of action regardless of the state, the speech lacks all credibility and, since those living in game theory land are not simpletons nor dupes, it is roundly and universally disbelieved.

As a logical analysis, the above is impeccable. As a description of leadership and persuasion, it seems to mis the boat completely. But, sadly, this analysis, or something similar, quite genuinely represents the state of the art, the research frontier, if you will. Is it fixable? Yes, in a way. We can add fools who believe everything the leader says into the mix. We can add some sort of inspiration variable that magically changes tastes so that followers work harder. But none of it really gets to the heart of what makes some leaders persuasive and others not. Indeed, we learn nothing if we simply assume that leader A can change minds and leader B cannot. The whole point of using our tools is to get at the deeper, and ultimately more interesting and important question as to why some leaders are persuasive.

Scenario #2: Factual Speech
Perhaps we've accorded too much freedom to our leader. After all, exagerrations, dissembling, misrepresenting, or any of the myriad of polite words we have for lying can get a politician into terrible trouble. Claiming that the world is hanging by a thread when, in fact, every poll shows that you're 20 points ahead, catches up to most leader's eventually. So let's return to our setting, precisely as before, but with the added restriction that our leader cannot simply make up things that are not true. In academic terms, this moves us out of the world of "cheap talk" and into the world of "persuasion" proper. This nomenclature, by the way, has a lot of problems. First, talk is no less cheap in the sense of being costless to the leader when we add the no lying restriction. Second, why the heck is it "persuasion" when we restrict someone from lying outright. Lies can be an important tool in the arsenal of a persuasive individual. Indeed, criminals engaging in confidence schemes are the ultimate persuaders, but would be entirely crippled were they bound by the no lying restriction. But I digress.

So let's rewind once again and send our leader back to the lectern, but with the following restriction--the heart of his speech can be either the truth or a stonewall, where he says nothing whatever about the state. One might imagine that this changes little. After all, when conflict was low, our leader did not wish to lie even when he could, so the restriction matters not a whit. When conflict was high, our leader wanted to lie about the state, but no one believed him anyway, so the effect is identical to stonewalling. Indeed, our leader in scenario #1 would have been quite happy to make the stonewalling speech instead of what I laid out.

In the case where conflict is low, the above supposition is exactly correct. Our leader steps to the lectern and offers a fact-laden speech truthfully revealing the state. But in the second case, this is wrong. Indeed, remarkably and perhaps absurdly, game theory offers the startling prediction that, no matter how bad the conflict between leader and follower, the leader always makes the truthful speech!

Why in blazes would our leader do that? Let's start with the situation where the threat is high. Here, the leader can do no better than to report the truth. He'd like more effort from his followers to be sure, but there is simply no way to motivate them to work any harder than by revealing the high state. What about the low state? Surely our leader will stonewall here? He might, but it will do no good since, knowing that the leader would have announced high were the state indeed high, our followers treat the stonewall speech as, in effect, a report that the state is low. And they act accordingly. That being the case, the leader might as well report honestly and at least gain the credit, however small, for being straight with followers.

Now, one may suspect that this logic takes too much advantage of the fact that there are exactly two states. What if there were three, or twenty, or a thousand. It turns out that none of it matters because of something called unravelling. Here's the argument: Suppose that there are twenty states in which the leader stonewalls while revealing in the rest. Then, in the highest of these 20 states, he'd be better off revealing than stonewalling since, by stonewalling, followers assume that the average state is lower than the highest state. Repeat this argument ad nauseum to obtain the truth-telling result. In my own work on the topic, I showed how this argument could be extended to virtually any configuration of preferences between leader and follower.

The problem is that the conclusion seems completely absurd. Irrespective of the conflict between leader and follower, the leader will always tell the truth sounds very much unlike the world in which I live. Again, this problem is fixable, but the main fix is even more bizarre than the result. It turns out that the key to credible stonewalling is...drumroll please...stupidity!! Or, more precisely the possibility of stupidity. The idea here is that, if the leader might possibly not know the state then stonewalling becomes believable. But this hardly seems like a satisfying resolution.

So Where Does This Leave Us?
This post, I'm afraid, is a downer. Game theory does lots of things well, but leadership, sadly, is not one of them. This has not stopped game theorists from trying, and perhaps making some headway. There is a clever paper by Dewan and Myatt looking at leadership through communications. In their model, one of the tradeoffs is between stupidity and incomprehensibility. They ask the following question (that only a game theorist would ask) about leaders: Is it better to be smart but incomprehensible or stupid but clear? The answer seems to be that it depends on the importance of doing the right thing versus doing the same thing. But, like all work in the field, the idea that leaders, with their words, could spark passion and devotion, is entirely missing.

Sometimes I despair about my love for game theory in a place devoted to, somehow, creating innovative leaders. I can, however, take some solace that we are no better at articulating how, exactly, that transformation takes place than we are in understanding leadership through game theory.

Thursday, October 9, 2014

The Limits of Experimentation and Big Data

Experimentation represents a critical tool for business decision making, especially in online environments. It is said that Google runs about a quarter of a million experiments on its users every day. Sophisticated online firms will have a built in toolset allowing managers to quickly and easily code in the experiment they wish to run (within certain limits), and even spit out standard analysis of the results. As a result of these experiments, firms continually innovate, mostly in small ways, to increase engagement, conversion rates, items purchased and so on.

The basic idea is simple. Firms with millions of visitors each day will choose a small percentage of these to be (unwitting) subjects for the experiment they wished to run. These individuals might be picked at random from all individuals or, more likely, they will be selected on the basis of some predetermined characteristics of individuals who are hypothesized to respond to the experimental treatment. In the simplest design, half of the selected subjects, chosen at random (or chosen at random within each strata) will receive the control, the experimental baseline, which will, most often, be simply business as usual. The other half will receive the treatment, which could be an alteration of the look and feel of the site, but might also be things like exposure to promotions, changed shipping terms, special offers of after sales service, or a host of other possibilities. Online sites rarely experiment with price, at least in this treatment-control way, owing to the bad publicity suffered by Amazon in the late 90s and late 00s from such experiments.

Following this, the data is analyzed by comparing various metrics under treatment and control. These might be things like the duration of engagement during the session in which the treatment occurred or the frequency or amount of sales during a session, which are fairly easy to measure. They might also be things like long-term loyalty or other time-series aspects of consumer behavior that are a bit more delicate. The crudest tests are nothing more than two sample t-tests with equal variances, but analysis can be far more sophisticated involving complicated regression stuctures containing many additional correlates besides the experimental treatment.

When the experiment indicates that the treatment is successful (or at least more successful than unsuccessful), these innovations are often adopted and incorporated into the user experience. Mostly, such innovations are small UX things like the color of the background or the size of the fonts used, but they are occasionally for big things as well like the amount of space to be devoted to advertising or even what information the user sees.

After all this time and all the successes that have been obtained, were we to add up the amounts of improvement in various metrics from all the experiments, we would conclude that consumers are spending in excess of 24 hours per day engaging with certain sites and that sales to others will exceed global wealth by a substantial amount. Obviously, the experiments, no matter how well done, are missing something important.

The answer, of course, is game theory, or at least the consideration of strategic responses by rivals, in assessing the effect of certain innovations.

At first blush, this answer seems odd and self-serving (the latter part is correct) in that I made no mention of other firms in any of the above. The experiments were purely about the relationship between a firm and its consumers/browsers/users/visitors etc. Since there are zillions of these users and since they are very unlikely to coordinate on their own, there seems little scope for game theory at all. Indeed, these problems look like classic decision problems. But while rivals are not involved in any of this directly, they are presently indirectly and strategically by affecting the next best use of a consumer's time or money, and changes to their site to improve engagement, lift, revenue and so on will be reflected in our own relationship to customers.

To understand this idea, it helps to get inside the mind of the consumer. When visiting site X, a consumer chooses X over some alternative Z. Perhaps the choice is conscious--the consumer has tried both X and Y, knows their features, and has found X to be better. Perhaps the choice is unconscious. The point is simply that the consumer has a choice. Let us now imagine the site X is experimenting between two user experiences, x and x' while firm Z presently offers experience z.. The consumer's action, y then depends not just on x but also on z or at least the perception of z. Thus, we predict some relationship
y = a + b x + c z + error

when presented with the control and a similar relationship but with x' replacing x under the treatment. If we then regress y on x, we suffer from omitted variable bias: z should have been in the regression but was not; however so long as z is uncorrelated with the treatment (and there is no reason it should be), then our regression coefficient on the treatment dummy will correctly tell us the change in y from the change in x, which is, of course, precisely what we want to know. 

Thus, buoyed by our experiment, we confidently implement x' since the statistics tell us it will raise y by 15% (say). 

But notice that this analysis is the statistical equivalent of inward thinking. Despite its scientific garb, it is no more valid an analysis than a strategic analysis hypothesizing that the rival will make no changes to its strategy regardless of what we might do. When we think about large decisions, like mergers, such a hypothesis is obviously silly. If Walmart acquired eBay tomorrow, no one would claim that Amazon would have no reaction whatever, that it would keep doing what it had been doing. It would, of course, react, and, were we representing Walmart, we would want to take that reaction into account when deciding how much to pay for eBay. 

But it is no less silly to think that a major business innovation undertaken by X will lead to no response from rivals either. To see the problem, imagine we were interested in long run consumer behavior in response to innovation x'. Our experiment tells us the effect of such a change, conditional on the rivals' strategies, but says nothing about the long-term effect once our rivals respond. To follow through with our example, suppose that switching to x' on a real rather than experimental basis will trigger a rival response that changes z to z'. Then the correct measure of the effect of our innovation on y is
Change in y = b(x' - x) + c(z' - z)

The expression divides readily into two terms. The first term represents the inward thinking effect. In a world  where others are not strategic, this measures the effect of the change in x. The second part represents the outward thinking strategic effect. This is the rival's reaction to the changed relationship that firm X has with its customer. No experiment can get at this term, no matter how large is the dataset. This failure is not a matter of insufficient power or the lack of metrics to measure y or even z, its the problem of identifying a counterfactual, z', that will only come to pass if X adopts the innovation. 

Now, all is not lost. There are many strategies one can use to forecast z', but one needs to be open to things that the data can never tell us, like the effect of a hypothetical rival reaction to a hypothetical innovation when viewed through the lens of consumer choice. This is not a problem that statistics or machine learning can ever solve. Game theory is not simply the best analysis for such situations, it is the only analysis available. 

Thursday, October 2, 2014

The Folk Theorem

We have talked a fair bit about coordination games and the forces shaping just what happens to be coordinated upon. In that light, it's important to realize how the presence of dynamic considerations, i.e. repeated game settings, fundamentally transforms essentially all games into coordination games. In our usual example to illustrate how to take advantage of dynamics to build relational contracts (self-enforcing repeated games equilibrium), we studied a simplified version of Bertrand competition. Firms could price high or low, low was a dominant strategy in the one-shot game, yet, with the right implicit contract, we can maintain high prices. The key, if you'll recall, was to balance a sufficient future punishment against the temptation to defect.

While coordinating on full cooperation seems the obvious course of action, if available, it is far from the only equilibrium to this game. For instance, it is fairly obvious that coordinating on the low price in every period is also an equilibrium--a really simple one, it turns out. Unlike the high price equilibrium with its nice and fight feedback loops, the no cooperation equilibrium requires no such machinery. The equilibrium consists of simply choosing a low price in every period regardless of past actions, and that's it.

There are many more equilibria in this game. For instance, if maintaining high prices yields each player $3 per period whereas fighting in every period yields $2 per period, then payoffs of any amount in between can also be sustained merely by interweaving the two. Suppose we wished to support payoffs that are high 5/6 of the time and low 1/6. In that case, we need only follow our high priced strategy whenever the period is not a multiple of 6 or if we're in a punishment phase and follow the low price strategy for periods that are a multiple of 6. Any fraction of high and low payoffs may be similarly supported.

This observation that just about any set of per period payoffs can be supported as an equilibrium is known as the Folk Theorem. It is so-named since, like many folk tales, no one was quite sure who first had the idea, but a Nobel winning game theorist, Robert Aumann, was the first to write the argument down (in a much more general and abstract way than my simple sketch above. Notice what the folk theorem implies in terms of our list of archetypal games:

All repeated games are coordination games
The theorem tells us that, regardless of the original form of the game, be it prisoner's dilemma, hawk-dove, matching pennies and so on, its repeated version amounts to a pure coordination game.

Sunday, September 28, 2014

Game Theory and Fleetwood Mac

Rumours by Fleetwood Mac is, in my opinion, one of the best rock albums ever released. The sales charts bear out this appraisal, Rumours, like Dark Side of the Moon, set records that may never be broken (especially with the advent of digital music) for album sales. But what has this got to do with game theory?

One song on Rumours, which became Bill Clinton's campaign theme song many years after, is Don't Stop Thinkin' About Tomorrow. The song offers the upbeat message that the future always holds something better than the past and so one ought to concentrate on its possibility rather than dwelling in the shortcomings of the day before. Perhaps useful advice for a heart broken teenager but hardly the stuff of deep insight. (Indeed, almost bitter advice for someone like me, whose joints and musculature are progressively being turned to Swiss cheese by an autoimmune illness.) But anyway, back to our story.

The broader point of this song is that the actions of the present ought properly to be considered in light of the future. In other words, moving back one step, one choosing how to act, one ought properly never to stop "thinkin' about tomorrow" since it is the consequences of tomorrow that determine the costs and benefits of today's actions.

And, indeed, no better or deeper point can be made about the most important insight in all of game theory that, by harnessing the future, the present may be tamed.

Rewind to our one-off prisoners dilemma situation. This situation seems utterly hopeless and, in the vacuum outside time, is hopeless. It makes not difference the timing of moves nor the sophistication of opponents, the inexorable conclusion in such situations is that both parties are doomed to defect and thereby receive the lower rather than thre higher payoff. But if we place this situation back in
time, where there is a future, then a more palatable (and sensible) conclusion obtains. So long as both
parties don't stop Thinkin' about tomorrow, and so long as tomorrow is important enough, cooperation is possible. Indeed, this simple insight is at the heart of the vast majority of "contracting" occurring in the world.

While the word contract brings to mind the formality of legal documents, its underlying idea is not so formal. A contract is any agreement willingly undertaken by two parties. While we may think of such informal contracts as arm's length "handshake agreements" they need not be. Spouses deciding on a rotation of chores or child care is no less a contract for its informality. Roughly speaking, anything that trades off a present benefit for one in either another form (money, cooking, etc.) or time (tomorrow, a year from now) is a contract.

What game theory says is that, even if such contracts hold no water in any court of law, they still might be fulfilled so long as they are "self-enforcing" which is a fancy way of saying that both sides find it in their interests to execute on the contract rather than renege. Much of game theory is casting
about for circumstances in which contracts

The key, in many instances, is that both sides don't stop thinkin' about tomorrow, which disciplines their behavior today. While defecting on a relational contract might be a fine idea today, when it is your turn to give up value/spend cost, such behavior seems less good in light of what is given up over many tomorrows where no one is willing to engage in future relational contracts.

Curiously, social psychologists independently discovered this idea, albeit without the help of game theory. They talk of "equity theory", the idea that we each keep a mental account of favors granted and favors received for each acquaintance. According to this theory, when the accounts fall too far out of balance, relationships are "liquidated" ---in effect declaring bankruptcy on the friendship.

The point though is really the same. If it is better not to honor a relational contract than to do so, such agreements cease to be self-enforcing and breach becomes inevitable. Where psychologists would
differ concerns favors never asked for in the first place. For instance, Ann is sick and so Bob makes her pots and pots of chicken soup, which Ann abhors and has never requested. To an economist, Bob's offering Places Ann under no particular obligation to repay whereas a psychologist (and certainly my grandmother) would see this as an odious debt accrued by Bob from Ann's care and attention.  Under the psych theory, Ann and Bob's relationship will likely founder over the unrequited chicken soup whereas an economist might see this as tangential, and indeed, irrelevant, to Ann and Bob's other dealings with one another.

Who do you think is corrcf? Our generic economist or my grandmother?

The IAWLP in auctions

The OPEC auctions are curious in a certain way.  While they are conducted a standard English auctions, they differ in that no one goes home without a prize (i.e. a country). Thus, even the loser of these auctions wins something. How does this change bidding?

We know from the IAWLP that the best strategy in a private values Vickey/English setting is simply to bid up to your value. But for a variety of reasons, our OPEC setting is not this setting at all. For one thing, values aren't really private--your estimate of the value of the auction is actually useful information for others seeking to determine the value.  For another, this is not a one-off auction. Losing a given auction implies that there are a number of other countries up for auction. Finally. There are only finitely many opportunities, so we can apply LFRB to anticipate how things might proceed. All of this makes the usual strategy of bidding "truthfully" whatever that means, a dead letter.

Let's start at the end, our usual game theory zen strategy. If there is only one auction left and only two bidders, how should you bid? Clearly, wat matters is not how much you value the UAE, but rather how much more you value it than Nigeria. This, of course, will depend on what you gleaned from earlier winning auction bids and bidders. The higher is the expected price over the course of the game, the larger the value of the gap between the production capacity of the UAE v Nigeria. At the same time, to achieve this level of production might require some curtailment of production on UAE v Nigeria that lessens this gap. None of this invalidates the usual strategy of bidding up to the point where you are indifferent between winning and losing, though it does affect where this indifference value lies.

So who wins this auction? Obviously, whoever thinks the UAE is worth more--whatever team is more confident that the price will be high and that the UAE can maintain high production while
maintaining this high price will get the item. Otherwise, it will be a toss up--the value will be the same to both players.

Now let's work our way back. The closer to the beginning of the auctions, the more moving parts on play. Early bidders need to worry about both the vagaries of valuations relative to the "numeraire" item, Nigeria, and in the face if ever declining future competition.  Both of these factors make these earlier bids more risky. But there is another wild card in the mix--leadership. Unlike beanie babies or even oil tracts, the items for bid here have values that are determined enormously by leadership activities. If you win Saudi, will you be able to convince others to refrain from producing and hence benefit? How much production will you yourself have to curtail to make these agreements work? In short, value is socially constructed by the winning bidder. In that sense, such auctions are neither private not publicly valued, rather the value of each country is interdependent--the winner of each country potentially determines the value of all.

In that respect, valuing a country in OPEC has, as its closest analog, valuing a company in a takeover. While the acquired company has a set of assets and ip which might be utilized, exactly how it is utilized and whether this is effective has everything to do with the acquirer. For example, the company Digg was bought by Yahoo some years ago. Digg was a very cool company at the forefront of crowd sourcing content and well ahead of its competitors. The synergies with Yahoo, a content curation company above all else, were obvious and important. The market raved over this acquisition.  But Yahoo treated Digg like it did all its other properties. Rather than integrating its technology to make for better curation, as the market anticipated, Digg was left to fend for itself as an independent revenue generating property, something it was never especially good at. As a result, it languished.

While I mention Digg to make a point, it is far from an isolated incidence. When valuing acquisitions, leadership in putting this asset to good use is central to this valuation Saudi in the hands of a poor leader is not a good bet at almost any (reasonable) price. In the hands of a master strategist, it is a bargain at almost any price. The game theory lesson (and it is a tough one to put into practice until you're near the top of the company) is that leadership plays a huge role in dictating the value of an acquisition, no matter what the cash flow or valuation multiple says.

Thursday, September 25, 2014

A/B Test

I am up in Seattle hanging out with the folks at Amazon today and talking about A/B tests, experiments that websites conduct in order to improve the user experience (and profitability too, sometimes). Coming home tonight, I hit upon the idea for a great Hollywood screenplay.

(Aside: I don't write screenplays, but, if any game theory alums do, I'd be happy to collaborate on this idea.)

Twins A and B (girls) have just come off bad breakups. They live in NYC and Boston but are otherwise alike in every way. They talk on the phone, as they do every night, feeling bad about their miserable love lives and determined to fix it. Twin A suggests they visit a popular dating website she read about. Immediately after the call, they each turn on their iPads and bring up the website. It turns out, however, that the site is conducting an A/B test on its matching algorithm just as they query it.

(Sidenote: as each twin calls up the website, a clock ticks down to the 1/1000 of a second as they query it. Twin A lands on an odd numbered time while twin B lands on an even numbered time since she started a tiny bit later than the other twin. This produces the A/B test, which is rigged to the 1/1000 of a second time a session starts.

Back scene: techies in some Silicon Valley startup. Techie 1 talks about how the website has succeeded by having likes attract. Techie 2 tells a story about how, with his girlfriend, opposites attract. What if that strategy actually produces better matches. Techie 1 says that data talks and bulllshit walks, so only an A/B test can settle things for sure. He proposes that, on Sunday night, they run such a test on the east coast and then track all the matches that result to see how love really works. Techie 2, confident that opposites attract, agrees and bets $100 that he's right. Techie 1 shakes hands on it.)

Back to our twins. Twin A is matched with someone just like her. He's outdoorsy and easygoing, ruggedly handsome and with a steady, albeit boring job. Twin B is matched with her opposite. She likes the outdoors while he is an urban creature favoring clubs, bars, jazz, etc. She is an all-American clean cut girl while he is rather grungy. After the first date, twin A has found her perfect match while Twin B is appalled, but yet fascinated somehow. Both go on subsequent dates and eventually fall in love.

The rest of the story traces out the arc of their lives. Since this is Hollywood, both have to run into terrible problems. If this is a PG film, then both will turn out to be with the right guys in the end. If rated R, then the person who seems to be alike will become a different, and controlling person, altogether. And violence will ensue, terrifying and possibly hurting Twin A.  The person who seems to be opposite of twin B will turn out to be alike in terms of his heart, so the outside bits don't count for much. He will also be the person to rescue twin A from her fate.

Or we can go rated R in the 70s. In this case, both guys turn out to be Mr. Wrong and wreck the twins lives. Divorced and with children to take care of, alone, the twins call each other in the ending scene to commiserate their fate.

The drawback to this idea is that it is a bit like Sliding Doors, a film from a few years ago, but different enough, I suspect, to be interesting.

No idea what this has to do with game theory, but it seemed interesting to me.

Thursday, September 18, 2014

The Paradox of Commitment

Commitment, in the form of renouncing or eliminating certain strategic possibilities ahead of time to gain an advantage, is one of the fundamental insights of game theory. Unlike decision problems, where more options can never leave one worse off> In interactive problems, games, where reactions come, not from nature, but rather from the exercise of free will on the part of others, more options, capabilities, capacity, client reach, and so on, is not always better. Thus, game theory offers numerous paradoxical situations where discarding or destroying assets, making certain strategies impossible to undertake and other such seemingly destructive behavior is, in fact, correct strategy.

One never, of course, does such things for one's own benefit. Rather, they are done to influence others playing the game to alter their course of action or reaction to a more favorable path. As such, these negative actions, the dismantling of a plant or the entering into of a binding contract removing a strategic possibility, must be done publicly. It must be observed and understood by others playing the game to have any effect.

One of the most famous situations illustrating the folly of commitment in private appears in the movie Dr. Strangelove where, in the climactic scene, it is revealed that the Russians have built a doomsday machine set to go off in the event of nuclear attack. Moreover, to complete the commitment, the Soviets have added a self-destruct mechanism to the device. It also goes off if tampered with or turned off. Since the machine will destroy the Earth, it ought properly to dissuade all countries to engage in nuclear combat.

But there's a problem--the US is only made aware of the machine after non-recallable bombers have been launched to deliver a devastating nuclear attack at the behest of a berserk air force commander. Why did the Soviets not tell the US and the world about the machine, asks Doctor Strangelove to the Soviet ambassador?
The premiere likes surprises. 
comes the pitiful answer. And so unobserved commitment is, in effect, no commitment at all.

While paradoxical initially, the idea that fewer options can improve one's strategic position is intuitive once grasped and was understood at an intuitive level long before the invention of game theory.

But I want to talk about a less well-known paradox: if such commitment strategies succeed by altering others' play in a more favorable direction from the perspective of the committing party, why would these others choose to observe the commitment in the first place? Shouldn't they commit not to observe in an effort to frustrate the commitment efforts of others? It turns out that this second level of commitment is unnecessary, at least in the formal argument, all that is needed is a small cost to observe the choices made by the committing party for the value of commitment to be nullified.

For example, two players are playing a WEGO game where they choose between two strategies, S and C (labels will become clear shortly). The equilibrium in this game turns out to be (C, C), but player 1 would gain an advantage if she could commit to strategy S, which would provoke S in response, and raise her payoff. Thus, strategy C can be thought of as the Cournot strategy while S represents the Stackelberg strategy in terms of archetypal quantity setting games. Suppose further that, if 1 could be sure that 2 played S, she would prefer to play C in response, so (S, S) is not an equilibrium in the WEGO game.

The usual way player 1 might go about achieving the necessary commitment is by moving first and choosing S. Player 2 moves second, chooses S in response, and lo and behold, IGOUGO beats WEGO as per our theorem. Player 2 is perhaps worse off, but player 1 has improved her lot by committing to S.

But now let us slightly complicate the game by paying attention not just to the transmission of information but also its receipt. After player 1 moves, player 2 can choose to pay an observation cost, c, to observe 1's choice perfectly. This cost is very small but, if not paid, nothing about 1's choice is revealed to 2. After deciding on whether to observe or not, player 2 then chooses between C and S and payoffs are determined.

Looking forward and reasoning back, consider the situation where 2 chooses to observe 1's move. In that case, she chooses S if player 1 chose S and C if player 1 chose C. So far, so good. If she does not observe, then she must guess what player 1 might have chosen. If she's sufficiently confident that player 1 has chosen S, then S is again the best choice. Otherwise, C is best.

So should she observe or not? If commitment is successful, then player 2 will anticipate that player 1 has chosen S. Knowing this, there is no point in observing since, in equilibrium, player 2 will choose the same action, S, regardless of whether she looks or not. Thus, the value of information is zero while the cost gathering and interpreting the information is not, so, behaving optimally, player 2 never observes and thereby economizes  (a little) by avoiding the cost c.

But then what should player 1 do? Anticipating that player 2 won't bother to observe her action, there is now no point in playing S since C was always the better choice. Thus, player 1 will choose C and it is now clear that the whole commitment posture was, in fact, mere stagecraft by player 1.

Of course, player 2 is no fool and will anticipate player 1's deception; therefore, if the equilibrium involves no observation, player 1 must have chosen C, and hence player 2 chooses C. Since we know that player 2 never pays the (wasteful) observation cost in equilibrium, the only equilibrium is (C, C), precisely as it was in the WEGO game. In other words, so long as there is any friction to observing player 1's choice, i.e. to receiving the information, first-mover commitment is impossible.

The issue would seem to be the conflict between players 1 and 2 where the latter has every incentive to frustrate the commitment efforts of the former since, if successful, 2 is worse off. But consider this game: Suppose that (S, S) yields each player a payoff of 3. (C, C), on the other hand, yields each player only 2. If player 2 chooses C in response to player 1's choice of S, both players earn zero while if the reverse occurs, player 2 chooses S and player 1 chooses C, then player 1 earns 4 while player 2 only earns 1. This fits our game above: C is a dominant strategy for player 1 while 2 prefers to match whatever 1 does.

This game has some of the flavor of a prisoner's dilemma. It is a one-sided trust game. By playing the WEGO game, both players lose out compared to the socially optimal (S, S), yet (S, S) is unsustainable because 1 will wish to cheat on any deal by selecting C. One-sidedness arises from that fact that, while player 1 can never be trusted to play S on her own initiative, player 2 can be trusted so long as he is confident about 1's choice of S.

Player 1 seeks to overcome her character flaw by moving first and committing to choose S, anticipating that 2 will follow suit. Surely now 2 will bother to observe if the costs are sufficiently low? Unfortunately, he will not. Under the virtuous (S, S) putative equilibrium, player 2 still has no reason to pay to observe player 1's anticipated first move since, again, 2 will choose the same action, S, regardless. Knowing this, 1 cannot resist the temptation to cheat and again we are back to (C, C) for the same reasons as above. Here the problem is that, to overcome temptation, 1 must be held to account by an observant player 2. But 2 sees no point in paying a cost merely to confirm what he already knows, so observation is impossible.

What is needed is a sort of double commitment--2 must first commit to observe, perhaps by prepaying the observation cost or by some other device. Only then can 1 commit to play S, and things play out nicely.

While paradoxical and logically correct, it seems quite silly to conclude that effective commitment is impossible. After all, people and firms do commit in various ways, their choices are observed, and these commitments have their intended effect. So what gives?

One answer is that, in reality-land, strategies are not so stark as simply S or C. There are many versions of S and likewise of C and the particular version chosen might depend on myriad environmental factors not directly observable to player 2. Now the information may be valuable enough that observation is optimal.

Seeds of doubt about the other's rationality can also fix the commitment problem. Ironically, this cure involves envisaging the possibility of a pathologically evil version of player 1. These evil types always act opportunistically by choosing C. Now there is a reason for 2 to look since she cannot be certain of 1's virtue.

A third possibility is that observing is simply unavoidable or that observation costs are negative. Curiosity is a trait common to many animals including humans. We derive joy from learning new things even if there is no direct economic value associated with this learning. Thus, individuals pay good money for intro texts on literary theory even though, for most of us, learning about Derrida's theories of literary deconstruction is of dubious economic value. Obviously, if the cost c were negative, i.e. a benefit, the problem vanishes and commitment is restored.

So if the theory is so implausible, then why bother bringing it up? One answer is to point out some countermeasures to commitment strategies. After all, if player 1 can "change the game" by committing to a strategy first, why can't player 2 change the game by committing to be on a boat in Tahoe and hence out of touch with what 1 is up to? A better answer is that it highlights the fact that commitment is a two-way street. Effective commitment requires not just that player 1 transmit the commitment information but that player 2 receive (and correctly decode) this information. Game theorists and others have spent endless hours thinking up different strategies for creating transmittable information, but precious little time thinking about its receipt. My own view is that this is a mistake since deafness on the part of other players destroys the value of commitment just as effectively as muteness on the part of the committing party.

Returning to Strangelove, it's not enough that the Soviet premiere transmit the information about the doomsday device ahead of time, for commitment to be effective such information must be heard and believed. This suggests the following alternative problem--even if the premiere had disclosed the existence of the doomsday machine, would the US have believed it? If not, Slim Pickens might still be waving his cowboy hat while sitting atop a nuclear bomb plummeting down to end all life. Yee-hah!