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You must obey the law, always, not only when they grab you by your special place.
You must obey the law, always, not only when they grab you by your special place.
[I wrote this essay with Yuanbo Wang.]
In How Robinson Crusoe managed his man Friday, I explored the interpersonal dynamics between two fictional characters, and concluded that these dynamics could be described as a game of power. Making reasonable assumptions about the characters’ preferences regarding their own the other person’s strategies (being domineering or docile), I drew up a payoff matrix and—using Brams’s (2011) Theory of Moves as a theoretical framework—recovered Defoe’s depiction of the two and their relationship. Crusoe and Friday got along, and Crusoe held on to the master’s power. In A general power game, I put the analysis in more abstract terms.
Tsebelis (1989) described the general power game years ago in his excellent paper in The American Political Science Review. His illustrative case was the power game between the police and the driving public. His Table 3 (p. 81) with the payoff matrix is copied on the left. The power game is characterized by the following differences between preferences. The public would rather have the police not enforce speed limits. But if there is enforcement, the public would rather not speed (c1 > a1); if there is no enforcement, however, the public would rather speed (b1 > d1). The police would rather have a docile public. But if they speed, the police would rather enforce (a2 > b2); if they do not speed, the police would rather not enforce (d2 > c2).
What is not explicit in Tsebelis’s exposition, and what I added in my analysis is the assumption that the public would prefer speeding under no enforcement to not speeding under enforcement (b1 > c1), and prefer not speeding under nonenforcement to speeding under enforcement (d1 > a1). This is the preference for tit-for-tat among the less powerful. Tsebelis also left implicit the assumption that the police would prefer nonenforcement if speeding to enforcement if speeding (d2 > a2), and prefer enforcement if not speeding to no enforcement if speeding (c2 > b2). This is the preference to counter with the opposite strategy typical of the more powerful.
This generic power game has no pure Nash equilibrium. Whatever combination of strategies might be encountered, one party has an incentive (can do better) by switching strategy. When the more powerful party (the police, Crusoe) switches from hard to soft (soft to hard), the less powerful party switches from soft to hard (hard to soft). Again, this pattern shows that the less powerful party can do well by playing tit for tat. In contrast, when the less powerful party switches from hard to soft (soft to hard), the more powerful party switches from hard to soft (soft to hard). Again, this shows that the more powerful party does well by not reciprocating the strategy of the less powerful party.
Following Luce & Raiffa (1957), Tsebelis provided the mixed-strategy Nash equilibria α and β for the public and the police, respectively.
α = (d2-c2)/(a2-b2+d2-c2) and
β = (b1-d1)/(b1-d1+c1-a1).
If the public (Friday) goes hard (easy) with probability α (1-α), the expected values of the two strategies are the same, and the player cannot be exploited. Likewise, if the police (Crusoe) go hard (easy) with probability β (1-β), they cannot be exploited.
Tsebelis notes that either player’s equilibrium probability only depends on the other player’s payoffs, and this is important. The police (Crusoe) can stiffen the penalty a1 on an onerous opponent, whereas the public (Friday) can do no such thing. Hence the power asymmetry. The surprising consequence—so Tsebelis – is that, though the police (Crusoe) can exert their power of punishment, they cannot change the public’s (Friday’s) behavior. Their probability of playing nice, α, remains unchanged. What changes is the police’s (Crusoe’s) own probability of playing hard, β, which goes down as a1 goes down (i.e., as the payoff of the less powerful for playing hard against a punishing opponent gets smaller or more negative).
A change in a1 is the most striking, most feasible, and most tempting intervention the more powerful party may attempt. Yet, any change in any of the payoffs for the less powerful party would only affect the probability of playing hard for the more powerful party. And the other way round. Whatever the less powerful party may do to change the payoffs of the more powerful party will only affect their probability of playing hard. The problem is, there isn’t much they can do anyway. Hence the power asymmetry.
On dérive
α: player 1’s probability of hard
β: player 2’s probability of hard
E1: player 1’s expected utilities
E2: player 2’s expected utilities
To calculate the Nash equilibrium probability for Player 1:
E2 = a2αβ + b2α(1-β) + c2(1-α)β + d2(1-α)(1-β)
= a2αβ + b2α – b2αβ + c2β – c2αβ + d2 – d2α – d2β + d2αβ
= (a2 – b2 – c2 + d2)αβ + (b2-d2)α + (c2-d2)β + d2.
To maximize E2:
dE2 / dβ = (a2 – b2 – c2 + d2)α + (c2 – d2) = 0
α = (d2 – c2)/(a2 – b2 – c2 + d2)
Similarly, we obtain the equilibrium probability for Player 2:
E1 = a1αβ + b1α(1-β) + c1(1-α)β + d1(1-α)(1-β)
= a1αβ + b1α – b1αβ + c1β – c1αβ + d1 – d1α – d1β + d1αβ
= (a1 – b1 – c1 + d1)αβ + (b1-d1)α + c1β - d1β + d1.
To maximize E1:
dE1 / dα = (a1 – b1 – c1 + d1) β + (b1 – d1) = 0
β = (b1 – d2)/(b1 + c1 – a1 – d1)
Tsebelis’s original assumptions (c1 > a1; b1 > d1; a2 > b2; d2 > c2) are sufficient for this analysis.
Brams, S. J. (2011). Game theory and the humanities. Cambridge, MA: MIT Press.
Luce, R. D., & Raiffa, H. (1957). Games and decisions. New York: John Wiley & Sons.
Tsebelis, G. (1989). The abuse of probability in political analysis: The Robinson Crusoe fallacy. The American Political Science Review, 83, 77-91.
