Erkenntnis

, Volume 77, Issue 2, pp 197–206

Identification in Games: Changing Places

Authors

    • Department of PhilosophyLingnan University
Original Article

DOI: 10.1007/s10670-011-9357-0

Cite this article as:
Rowbottom, D.P. Erkenn (2012) 77: 197. doi:10.1007/s10670-011-9357-0
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Abstract

This paper offers a novel ‘changing places’ account of identification in games, where the consequences of role swapping are crucial. First, it illustrates how such an account is consistent with the view, in classical game theory, that only outcomes (and not pathways) are significant. Second, it argues that this account is superior to the ‘pooled resources’ alternative when it comes to dealing with some situations in which many players identify. Third, it shows how such a ‘changing places’ account can be used in games where some of the players identify with one another, but others do not. Finally, it illustrates how the model can handle the notion that identification comes in degrees.

1 Identification and the Prisoner’s Dilemma

Consider a simple Prisoner’s Dilemma scenario, as represented in Fig. 1. Classical game theory suggests that either player ought to select defect, although empirical studies indicate that many people—in fact, only marginally under 50% on the average over the studies considered by Sally (1995)—select cooperate.1 So rather than accept the implausible consequence that a significant percentage of people are irrational, authors such as Bacharach (2006) invite us to conclude instead that some form of identification is occurring between the players.
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Fig. 1

Simple prisoner’s dilemma

The purpose of this paper is not to explain why identification occurs—that is, why individuals prefer cooperative strategies in some circumstances—but to propose a new way to understand, or model, the mechanism by which it occurs. So in the subsequent discussion, we will take it as given that identification occurs for reasons independent of the games under consideration.

The first hurdle is to decide whether it matters how a game is played. Identification might, for instance, merely consist of empathy with respect to the experience of being cheated or defeated in some game; the disutility, that is to say, of being on the receiving end of ungenerous, or downright selfish, behaviour. Alternatively, identification might be concerned purely with the outcome. I take this to be preferable because the model is kept simple; we need not worry about pathways to consequences rather than consequences themselves. As we will see, however, there is more than one model of identification that respects this requirement.

Now my basic idea is that we can model identification, say of player one with player two in Fig. 1, by imagining that player one understands that he might find himself in player two’s shoes with respect to the outcome. So we can imagine him knowing that he will make the choice of cooperate or defect as player one, but that he will then have a half chance of swapping places with the other player (i.e. becoming player two) at the conclusion of the game (Fig. 2).
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Fig. 2

‘Changing places’ perspective of one player identifying with another

Intuitively, opting for defect is now the less attractive option (even if the other player sees the potential results differently).2 But is the probability supposed to be understood as aleatory, i.e. to reflect a propensity or a statistical frequency? Or is it supposed to be understood as personal, e.g. as a coherent degree of belief? Taking inspiration from the idea of the veil of ignorance, in the version of Vickrey (1945) and Harsanyi (1953, 1955), I opt for the latter.3 To identify with someone else, in the sense proposed here, is to behave as if one might end up in their shoes (and they might end up in one’s own shoes). Being given the possibility of ending up in any position in society, and assigning each possible outcome an equal probability, leads one to consider carefully how best society should be organised for its citizens. Similarly, I contend, considering a game from such a perspective leads one to consider how best the game should be conducted for its participants.

This raises the question of whether it is rationally required to assign each possible outcome an equal probability, e.g. to follow the principle of indifference advocated by Keynes (1921) or the modern equivalent, the maximum entropy principle, advocated by Jaynes (2003) and Williamson (2010). The basic idea behind such principles is that one should equivocate over possibilities in the absence of any information to indicate that any particular possibility, rather than any other, obtains (or will obtain); and the difficulties with this idea have been discussed extensively, e.g. by Seidenfeld (1986), Gillies (2000), and Rowbottom (2008, 2011).

However, it is not necessary for such a rule to hold in order for the ‘changing places’ account to be tenable. Instead, it may be proposed simply that (full) identification involves assigning equiprobability across possible place changes (and the possibility of not changing places). So in an n + 1 player game where a given player fully identifies with the other n players, his ‘changing places’ probability (or swap probability) for each other player will be 1/n.4 And if that condition fails to obtain, we must conclude that the given player does not fully identify with the other n players. As we will later see, this opens the door for modelling degrees of identification too. (These seem to exist given that it is intuitively plausible to say that one might identify rather more with one player than another, yet still identify with both to some extent, and so forth.) In essence, the idea is that swap probabilities may differ. But for the moment, until Sect. 4, take all mentions of ‘identification’ to refer to full identification.

2 ‘Changing Places’ Versus ‘Pooled Resources’ Approaches

Naturally, there are alternatives to the ‘changing places’ approach outlined above. One notable option is to take what might be called a ‘pooled resources’ perspective. Here, as depicted in Fig. 3, the player identifying with the other would be said to imagine that the overall utility from the outcome will be shared equally.
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Fig. 3

‘Pooled resources’ perspective of one player identifying with another5

Again, the result appears to be intuitively desirable. Clearly cooperate is the best option for the player with a perspective such as that outlined in Fig. 3. However, finding a justification for modelling typical situations of identification in such a fashion is not easy; there is no obvious analogue of the veil of ignorance in the ‘changing places’ model. Moreover, the pooled resources view leads to undesirable results, as we shall see below, when somewhat more complicated games are considered.

Consider now a three-way scenario akin to the classic Prisoner’s Dilemma, as depicted in Fig. 4. With no identification present, the best option for any player is clearly to defect rather than cooperate.
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Fig. 4

Three player Prisoner’s Dilemma scenario

Think now, however, about how the situation looks from the perspective of a player who identifies with the other two on the ‘pooled resources’ model of identification (Fig. 5).
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Fig. 5

Three player Prisoner’s Dilemma scenario for a ‘pooled resources’ identifier

Here, cooperation is no more appealing than defection. (Imagine player three is the ‘pooled resources’ identifier. Cooperation gives two chances for 2 and two chances for 1; defection is indistinguishable.) But on a ‘changing places’ perspective of identification as depicted in Fig. 6, by way of contrast, we still achieve the desirable result that cooperation is the more attractive option provided that the player is risk averse (i.e. she would rather have ‘2 in the hand’ than a third of a chance of ending up with nothing and two-thirds of a chance of ending up with 3).
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Fig. 6

Three player Prisoner’s Dilemma scenario for a ‘changing places’ identifier6

Admittedly, we would like not to have to assume that the identifying player is risk averse; however, this model still trumps the ‘pooled resources’ one because in that event all the results are ‘in the hand’.

3 Identification of Sub-Groups on a ‘Changing Places’ Model

The fruits of the ‘changing places’ model of identification may also be seen by considering a scenario in which players two and three identify with one another, but neither identifies with player one. This is a version of the Three Player Prisoner’s Dilemma where two of the prisoners are partners in crime, but the third stands alone (Fig. 7). (Perhaps he is not part of the gang that the other two are in. Maybe he is a driver that was hired for the botched bank job that resulted in the arrest of the three players.)
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Fig. 7

Three player Prisoner’s Dilemma scenario for a ‘changing places’ identification between players two and three (and/or vice versa)

As indicated, the best option for two and three is to both defect; that way, they are both protected from losing out altogether (and also have the prospect of both gaining as much as the game allows in principle, in the event that player one decides, foolishly, to cooperate). This is the result that we want.7

It is true that a similar result can be gained from the ‘pooled resources’ model of identification—see Fig. 8—but we have already seen why this model is inferior in other circumstances.
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Fig. 8

Three player Prisoner’s Dilemma scenario for a ‘pooled resources’ identification between players two and three (and/or vice versa)

4 Extension of the ‘Changing Places’ Model to Cover Degrees of Identification

A further advantage of the ‘changing places’ model, which was briefly mentioned in the introduction, is that it provides a natural means by which to cope with the notion that identification comes in degrees. Consider first a simple two player Prisoner’s Dilemma from player one’s perspective (Fig. 9).
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Fig. 9

Possible degrees of identification of player one with player two in a Prisoner’s Dilemma

If n is one half, then full identification occurs. If n is zero, then no identification occurs. And if n is greater than zero but less than one half, then partial identification occurs.8 (What if n is higher than one half? We will come to this shortly.) Thus there is a specific range of degrees of identification for which defect is a preferable option to cooperate (and vice versa). One may consider an expected utility calculation in order to see this. Imagine, for instance, that player one thinks that player two is equally as likely to cooperate as to defect; P(two cooperates) = P(two defects) = 0.5. The expected utility of selecting cooperate is 0.5 × 2 + 0.5 × 3n, whereas the expected utility of selecting defect is 0.5 × 3(1 − n) + 0.5 × 1, from player one’s perspective. Setting these equal, we can find the point at which both selections have the same expected utility:
$$ \begin{aligned} 1 + 3n/2 & = 3/2 - 3n/2 + 1/2 \\ 3n & = 1 \\ \end{aligned}$$

Hence when n is greater than a third and less than one half, cooperate is the preferable option for player one; when n is lower than a third, defect is preferable.

It is also possible to allow n to be larger than one half but less than or equal to one, in which case we might say that player one overidentifies with player two. In the extreme case where n is one, for example, player one is concerned only with how the outcome of the game will affect player two. He will therefore see cooperate as the preferable choice. (One might think it more accurate to say that full identification occurs when n is one, and that player one equally identifies with himself and player two when n is a half; that is, if one allows for self-identification. However, I prefer to adopt the current convention for ‘identification’ in order to avoid confusion.) Overidentification need not be irrational, although it may often be altruistic and supererogatory.9

The advantages of allowing for degrees of identification become even clearer when we consider games involving more than two players. To see this, let us again consider a Three Player Prisoner’s Dilemma (Fig. 10).
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Fig. 10

Player one’s degrees of identification in a three player Prisoner’s Dilemma

If n has the same value as m, then player one fully identifies with player two; and if 1−mn also takes the same value as n, then player one fully identifies with player three too. But player one may fully identify with player two while identifying with player three to a lesser extent, for example, in the event that m = n > 1−mn. Then if the difference in degree of identification is great enough, he will prefer a different option from the one he would if he fully identified with both players. (Naturally, other orderings of identification are possible: n > m > 1−mn; n > 1−mn > m; n = 1−mn > m; and so forth.) In general, it is possible to perform expected utility calculations, such as the one illustrated above, in order to determine which option player one should prefer for any given values of n and m.

Before concluding, there are two related questions that I should like to address. First, are there any reasons independent of the prior considerations to think that identification genuinely involves considering changing places? (And does it matter if it really does?) Second, is the ‘changing places’ approach superior to the ‘pooled resources’ approach in general, i.e. in all possible applications?

5 The Reality and Scope of the ‘Changing Places’ Account

There may be some reason, independent of the prior considerations, to think that people genuinely ‘put themselves in the shoes of others’ when they’re considering how they should behave. For example, one might appeal to the evidence concerning the existence of mirror neurons in humans—see Iacoboni et al. (1999)—and the simulation theory of mind championed by Goldman (2006). And although it is not plausible that people make rather complicated expected utility calculations in order to decide what to do, the normative aspect of game theoretical approaches must be borne in mind. In short, a theory of what one ought to do when one identifies may be correct in spite of what people actually do.

As an anti-realist, however, I think that whether the ‘changing places’ model is realistic is of little concern; I am interested in whether it saves the phenomena, i.e. gives good predictions, in an economical way.10 So I commend it, on pragmatic grounds, as more theoretically virtuous than the available alternative (in, as I will explain below, a significant class of scenarios). In particular—to borrow from Kuhn’s (1977) list of theoretical virtues—I have argued that it has more accuracy and scope than the ‘pooled resources’ approach, although it is somewhat less simple. (There is often a trade-off to be made between scope and simplicity; by introducing further complexity into one’s models, one can handle more scenarios. There are many examples in elementary physics, e.g. in the move from the ideal gas law to the van der Waal’s gas law.11) So in summary, I would say that the ‘changing places’ model is a useful way of thinking about identification; it is a convenient fiction.

This brings me to the question of whether I have shown that the ‘changing places’ approach is generally superior to its ‘pooled resources’ rival. I have not, of course, because I have not had the luxury of considering all possible identification-based scenarios. Thus I concede that there may be some circumstances in which it is easier to use the latter than the former. (An analogy, given what I have said above, may help. No-one uses special relativity, rather than Newtonian mechanics, when doing biomechanics; the former is easier to use in such scenarios, although it is less accurate and has less scope.) I also concede that there may be some circumstances in which it is possible to use the latter but not the former—when unusual internal group structures are present, for example—although I cannot presently conceive of any.

6 Conclusion

We have considered two different ways of modelling identification, via ‘changing places’ or ‘pooled resources’ approaches, and seen that the former is superior on a number of counts. (It should also be remembered that ‘team perspectives’ are similar to ‘pooled resources’ approaches; see footnote five.) It may be employed to favour cooperation (when risk aversion is present) in three (or more) player Prisoner’s Dilemma scenarios with full identification, and also be used to explain the preferences of identifying sub-groups in such games. Furthermore, crucially, it may also handle the notion that identification comes in degrees, and therefore that any given player in a game may identify with several other players to some extent, but with some more than others. And all this is achieved without abandoning the assumption that identification may be understood in terms only of the outcomes of (and not the pathways that occur in) games.

Footnotes
1

Fear of reprisal is ruled out in studies where neither player knows the identity of the other, and both know that they will only play the game with one another once.

 
2

I will not, here, try to fully explain that intuition; this is a topic explored by Sugden (1993, 2000) and Bacharach (2001, 2006).

 
3

Rawls (1971), who is perhaps most often associated with the ‘veil of ignorance’ idea, wanted to avoid the use of probabilities in decision making; instead, he proposed to use a maximin decision rule. This is explained and criticised by Harsanyi (1975).

 
4

A game with finitely many players is assumed throughout; the model will break down for infinitely many players, because zero probabilities may nonetheless correspond to possibilities in such circumstances. See, for example, Williamson (2007).

 
5

Taking a ‘team perspective’, i.e. considering combined utility, results in a similar ranking. The payoff at the level of the team is 4 if both players cooperate, only 2 if both players defect, and 3 if one player defects and the other cooperates.

 
6

I should mention another possible model, where each possible change of place is taken into account, and multiple swaps are allowed. In the three player case, for example, one would consider six possible orderings: 1, 2, 3; 2, 1, 3; 3, 2, 1; 1, 3, 2; 2, 3, 1; and 3, 1, 2. Clearly, such a model is considerably more complex than the one considered here. It is conceivable, however, that it may be superior in some situations.

 
7

Identification need not be taken to be reciprocal in the account offered here. We can also imagine a case where three identifies with two but two does not identify with three; then three considers the situation as depicted in Fig. 7, but one and two consider the situation as depicted in Fig. 4 (provided neither identifies with the other).

 
8

One need not think that there are precise numerical degrees of identification (just as one need not think that there are precise numerical degrees of belief); in principle, one may work with imprecise probabilities instead.

 
9

Modifications to the ‘pooled resources’ approach might also be made in order to allow for degrees of identification; for example, a given player might assign weights to the utilities of the other players. The use of probabilities in the ‘changing places’ account makes the development natural, however, in a way that it has not previously been.

 
10

See Rowbottom (in press).

 
11

In the latter, but not in the former, the volume of molecules and the forces between molecules are taken into account.

 

Acknowledgments

I am grateful to three anonymous referees for their comments on a previous version of this paper, which helped my thinking on these matters considerably. I should also like to thank the British Academy and the Sydney Centre for the Foundations of Science for supporting this research.

Copyright information

© Springer Science+Business Media B.V. 2012