But it’s so simple. All I have to do is divine from what I know of you: are you the sort of man who would put the poison into his own goblet or his enemy’s? Now, a clever man would put the poison into his own goblet, because he would know that only a great fool would reach for what he was given. I am not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool, you would have counted on it, so I can clearly not choose the wine in front of me. – Vizzini (The Princess Bride)
Contrary to initial expectations, we’ve found that a perfectly rational player ought not to defect in a game described in Part I. So what? How does that apply to the real world? Game theory is useful for modeling decision making processes where abstractions reduce the variables to be considered. Examples in real life of situations modeled by a tragedy of the commons dilemma are readily available.
Real world behaviour deviates from what models predict for perfectly rational (or superrational) individuals for at least two reasons: though selfish, many individuals are not perfectly rational or superrational and factors abstracted out in the model play a role in decision making. In situations where optimal behaviour dictates that all players should cooperate, individuals that defect place an undue burden on superrational players and rational players that cooperate ((Note that rational players are a subset of perfectly rational players.)). In some cases, such players may cause all players they interact with to fail. Below are some real-life tragedy of the commons situations to which I apply the logic seen in Part III. I shy away from value judgements; only impacts on efficiency and individual utility are examined. I will therefore avoid things like ethics that are necessary to make value judgements. We will see how game theory can be applied to vaccinations, modeling employee work ethic, and reducing greenhouse gas emissions and whether behaviour that is only perfectly rational dooms its players to be in a race for the bottom as J would predict.
Vaccinations have two benefits: the prevention of the spread of communicable diseases and prevention of infection to the vaccinated individual ((We will assume for the purposes of discussion in this post that all people highly value their continued existence and well-being and that each person does so to the same degree; clearly, this is not true in reality, as evidenced by the existence of suicidal individuals.)). The former is a social benefit and the latter an individual benefit ((We will ignore the benefit of not transmitting a disease to someone you care about. The analysis does not significantly change if we take this into account, but it does come into play when looking at diseases that affect different subsets of the population differently. However, the end conclusion by a superrational person is the same.)).
We will assume that the benefits of obtaining a vaccine is greater than the cost of getting the vaccine. The cost of a vaccine is time, money, and possible side effects of a vaccine. If everyone gets vaccinated except one, the lone defector obtains the benefits without the costs by freeloading. In fact, the perfectly rational individual will defect (not get vaccinated) because the game-theoretic perfectly rational individual does not take into account the behaviour of others. Further, unlike J’s game, there is no übergroup — the “competitors” in the game appear to be every other person. However, defection is not usually in the average person’s best interests. Indeed, vaccinations in a world only full of perfectly rational individuals would be a situation that results in a race for the bottom since nobody gets vaccinated. That would clearly be a problem. Instead, we would hope there are superrational individuals (and non-superrational individuals that choose to be vaccinated).
The superrational individual would reason somewhat along these lines: If I were not to receive a vaccination, I would not incur the costs of getting a vaccine. However, if I am surrounded by superrational (or perfectly rational individuals) that follow this line of reasoning, I will not get the social benefit provided by others getting vaccinated. Therefore, to prevent myself from getting sick, I ought to get vaccinated ((We have assumed that the costs are outweighed by the benefits — otherwise, the vaccine would not have been developed.)). Thus, all superrational individuals (who are, by definition, selfish) would get vaccinated against any nasty diseases.
Verdict: A world of only perfect rational agents results in a very sick populace.
Employee Work Ethic
If you ask most people why they are employed, the answer would probably be to make a living. You might get a few oddballs, like academics, that actually enjoy what they do. If you probe further, you will find that most people value free time. Thus, we can consider the utility of working at a job (U) to be Value of Earned Money (VEM) + Value of Job Enjoyment (VJE) – Value of Free Time (VFT). In this simplistic model, one expects cooperation (work) so long as VEM+VJE > VFT. That is, one continues to work so long as the benefits outweigh the benefits of having free time. In this section of the blog post, we will examine only the effects of VEM, VJE, and VFT on productivity.
The World of Wally
This simplistic model works when applied to Wally, a character in the comic strip Dilbert who is notorious for his ability and desire to do no work and get paid. In the dysfunctional company that he works for, salary is not linked to performance and he suffers no ill effects for his efforts, or lack thereof. Wally generally appears to be happy with his wage, unlike some of his coworkers, and therefore has no incentive to change the status quo. Therefore, we see that Wally places less utility on any additional salary than he does on his free time.
The above example illustrates the first wrench in the works in attempting to naïvely map this situation onto J’s game: the marginal utility of VFT and VEM (we will ignore any boredom from doing a job that one enjoys). In the case of Wally, he earns enough, and places little value on extra money, so he has almost maximized VEM; all that is left for him to do is to maximize his free time to maximize VFT. To see how marginal utility affects U for a worker in general, imagine someone who has no money and needs to make ends meet. We would expect that this person would place a high value on a job that just meets his/her living expenses. But, due to diminishing returns on utility (marginal utility), a person that has more money in the bank places less value on the same job, all else equal to the extent that the richest person in the world is unlikely to take the job. The marginal utility of VEM can be seen when looking at retirees: with so much free time on their hands, some know not what to do and return to work. Therefore, it would be more appropriate to evaluate VFT and VEM at various levels of cooperation: U(c) = VEM(c) + VJE – VFT(c). This explains why we would not expect to find only workaholics and completely idle employees in the workplace: each employee cooperates to the degree that maximizes U(c).
The Real World
In the real world, one’s true competitors are not people in the same company, but the company’s competitors (the übergame in J’s game). What is bad for one’s company is bad for the individual. In a real company ((We necessarily have to ignore the government for this part of the exercise, but you are free to apply the rest of the logic to civil servants.)), if everyone were like Wally and did nothing all day, the business would fail unless all competitors were equally defective; due to layoffs, monetary compensation drops to zero (i.e., VEM becomes zero), while employees at other companies continue to draw salaries ((Unlike in J’s game, one could try applying for a job with a former competitor, although poor performance at a previous company might make actually landing the job difficult.)). At the other extreme, if everyone worked extremely hard, all other factors being equal for competitors, the efficiency of the company would increase, resulting in greater profits for the company which can then afford to reward its employees.
In cases where performance has little effect on compensation, as in the example of Wally, VEM(c) is can be treated as a constant: regardless of the amount of work done, compensation remains the same, so VEM(c) is the same. Employees that do not think perfectly rationally or superrationally do not see that idleness (in a workplace full of some mix of perfectly rational individuals and always-defectors) causes them to risk a VEM of zero. Why would a (perfectly) rational individual keep working when he/she realizes the company won’t have enough money to give out paycheques at the end of the month ((Note that backwards induction does not work here for the same reason it didn’t work in J’s game: If you won’t get fired in the last second before you quit/retire from not doing work because all other “perfectly rational” (according to J) players will also defect, why would you work for the second last second? Why would you work at all?))?
On the flip side, employee compensation is closely linked to company performance when employees have a stake in the company or there is a form of profit sharing. Less direct ways such as performance bonuses have a similar effect. When employees defect (do nothing), they gain utility (free time) — and if they are laid off because everyone else is defecting, they have the same utility (free time) minus the financial results of others’ contributions; this is akin to J’s game. Because of this, in a real company, one will likely find a mix of Wallies and hard-workers. As long as the hard-workers prevent the company from going under and the Wallies are allowed to keep their jobs, the company is not operating very efficiently.
What we see, then, is that in a company that rewards good performance, as long as perfectly rational employees are being compensated for work more than they value their free time, they will cooperate. The debeakered model of J’s game would suggest that a stronger connection between the performance of the company and the compensation of the employee will lead to maximal cooperation (effort) ((Note that this is not the same as obtaining full cooperation: because employees still place value on VFT, there will be periods that an employee is not working for the company.)). One company that uses this model successfully is WestJet ((This is of course a biased sample of one and is used merely for illustrative purposes.)). However, even in cases of self-employment where performance is most closely tied to compensation, in many societies, there are structures in place that reduce the link between performance and compensation.
Some welfare systems (including Milton Friedman’s negative income tax) cause a discrepancy in compensation relative to the amount/type of work performed. When money for welfare is freely available to the unemployed, individuals that can then cover their living expenses and highly value their free time have no incentive to work because of a lower-bound on income. Further, money for welfare needs to come from some place. Unless the welfare is completely funded by voluntary charitable acts, it is levied on a subset of the populace unwillingly. When tax on income is introduced (we will ignore forms of income not tied to a job such as dividends and interest), this has the effect of reducing the amount of work a person is compensated for while the work imposes the same burden on free time. With progressive taxes, the degree of income “flattening” is most pronounced and thus provides the strongest disincentive to work. A flat tax by itself has little effect on VEM, though welfare itself has a much stronger one. A regressive tax creates the most incentive to work as it partially offsets decreases in desire to work due to the marginal utility of VEM.
The existence of labour laws reduce the efficiency of companies. In the absence of legal protection, a union’s power is theoretically proportional to the supply and demand of labour. When labour is plentiful, if a union’s demands are too great, a company would be able to simply fire all unionized workers and rehire new ones from the overall supply if necessary. In times of labour scarcity, a company would acquiesce to a union’s demands until the cost to the company of meeting the union’s terms outweigh the utility provided to the company. Instead, because of laws in place, as witnessed during the recent GM union negotiations, unions whose agreements are protected by labour laws can exact an unsustainable burden on the hand that feeds it, even with the knowledge that its actions will ultimately lead to its own demise! Unions (and in some ways tenure) that protect idle workers remove the incentive to contribute to the company, promoting “defection” so long as there are a sufficient number of “cooperators” to keep the company afloat.
In some places, workers are entitled to a minimum wage. This results in market inefficiencies, resource allocation inefficiencies, and a loss of incentive to work. A lower bound on minimum wage has the effect of creating an artificial scarcity in the labour market: fewer employees can now be hired since a company must now expend more money for a job an unregulated market could have provided for less and the company has to pay more for each employee ((From a market inefficiency perspective, instituting minimum wage laws have the same effect as cornering the market, price-fixing by cartels, and the sale of “limited edition” items.)). While market inefficiencies do not have a direct effect on VEM, VJE, or VFT, the resource allocation inefficiency means cutbacks elsewhere are required. The manner in which cutbacks occur have effects on VEM that mirror taxation. The loss of incentive to work is a result of flattening the VEM function as with welfare by putting an artificial lower bound on income.
Verdict: A world of only perfect rational agents (with no unions or labour laws) results in a world with no workers that always idle.
Greenhouse Gas Emissions
Many politicians are discussing reducing greenhouse gas (GHG) emissions, but making precious little action. Unlike J’s game, players do not take turns simultaneously. Instead, each player announces its intentions publicly whenever it likes and sometimes in advance to other players through back-channels. Further, players can also backpedal at any time after cooperating. Unless one can identify an übergame, this becomes a single group version of J’s game where defection is the dominant strategy, though with some added twists. Each country thus corresponds to a player. Unlike in J’s game, not only do the cost of reducing emissions and ability to do so vary with each player, so do the benefits to each, whether we take into account climate change or not.
Supposing we have agreed on the amount by which we need/want to reduce our GHG emissions (or even remove the existing GHGs from the atmosphere), unless any “player” can cut GHG emissions sufficiently (and with no chance for other players to increase theirs to upset the first player’s plans), since the initial state of the game had insufficient cooperators, if all world leaders are only perfectly rational, they will continue to defect. Even if one player tries to cooperates, if no one follows suit, if the player is rational, it will then switch back to defect.
If only superrational players are playing, there might be sufficient cooperation amongst nations to cut GHG emissions to keep our planet’s climate in check since the costs of even a four degree rise in mean global temperature is predicted to have huge implications for humans and the earth. The only reason I add “might” for superrational players is that some places, like the geographical areas currently occupied by Canada and Siberia, would benefit, and thus have a motivation to induce climate change — although the possibility of high immigration, wars, and a devastated world economy would mean Canada and Siberia as we know them would no longer exist ((Unless the present nation of Canada could survive or the individuals governing the nation think that they would also control a successor state, even though the region may benefit from climate change, those in power now should seek to prevent climate change to protect their own interests.)). Again, we assume that one’s utility from retaining political power pales in comparison to survival. Also, any individuals who do not care what happens in the timeframe of accelerated climate change (possibly because they do not expect to live that long), even if superrational, would not cooperate. If we exclude the last scenario, even if the players are superrational and loss of control of a country is the same as a player loss (the agent formerly making decisions is no longer making decisions), we will possibly still have some “defecting” countries that raise emissions. A superrational player would maximize world utility but, because of differences in costs and benefits to each region, some countries may still be permitted to marginally increase GHG production, so long as long as overall production drops by a sufficient amount to keep climate change in check ((If climate change is not kept in check, all the superrational players incur a huge negative utility from lack of survival, so, naturally, superrational players will avoid this situation.)) and those countries obtain a larger increase in utility than is lost by the other countries from reducing their emissions by the same amount. This probably corresponds to a plan for current large emitters of GHG (per capita) to sharply curb emissions while still allowing for some growth in countries with a small footprint.
Okay, so maybe it is too much to hope for to have each (or even most) countries led by a superrational individual. Does that mean GHG emissions will continue to rise unchecked? No. In the game described, cooperation is expected to be monotone decreasing only in the absence of communication and independent decision making processes. One approach to cutting GHG emissions globally is to use the former point to our advantage and do what Canada is doing: tie one’s carbon emission policy to the United States. Even if the Harper government is doing this for the wrong reasons (expecting the US to “defect” or make only a token gesture), if a sufficient number of countries do this, the US, assuming it is perfectly rational or superrational (manifest destiny notwithstanding), would switch its strategy to “cooperate”. Due to the linked policies, the US’s cooperation would effect cooperation in the other countries, producing the desired effect of survival of the species. But with the survival of humans at stake, can we really afford to be playing games?
Verdict: A world of only perfect rational agents results will test the dangerous scenarios of climate models.
Logic from the simple game presented by J can be used to determine the most logical course of action of a perfectly rational or superrational player. We see that in some cases, though the natural instinct in many situations may be to defect, cooperating can still be in one’s best interests, but it may require superrationality or external forces to break the cycles of defection. Even with the framing held constant, context can change the way logic is applied, as Wason and Shapiro showed in an very different context using Wason’s selection task. It is interesting to compare what the simple model described in this 4-part series predicts a perfectly rational or superrational individual will do to what happens in real life. I’ve only taken a brief look at three things above (and, believe it or not, edited it for brevity), but there are many more examples such as deciding whether or not to take public transit, to vote in an election, to volunteer to coach a little league team, and to put things in the bin. When looking at different situations as above, it is somewhat reassuring to me that selfishness can often induce cooperation, whether the stakes are large or small. Perhaps we all need to take Vizzini’s advice to heart: Never go against a Sicilian when death is on the line.