The consequence and support graphs
The equilibrium was set forth as paths of play, but another mode of description fits the systemofnorms interpretation somewhat better (O’Neill 1999). One can construct the equilibrium’s consequence graph and support graph, Figure 1. Each has nodes for the five states that current play might be in. Each state has an associated norm, instructing both players what to do. The consequence graph indicates where play will go at the next state as a consequence of compliance with or violation of with the current norm. The support graph (which is constructed immediately from the consequence graphs), shows, for the norm associated with a given state, which norms at various states support it, through the player’s expectation of rewards or punishments.
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FIGURE 1 ABOUT HERE
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The graphs in Figure 1 take Row’s viewpoint, and corresponding versions exist for Column. Consider Row’s consequence graph (Figure 1, top.) If play is at the leftmost oval, the state of mutual cooperation, Row is to play Transfer. If Row does so, the game stays in that state, as indicated by the solid arrow. If Row violates the norm by choosing W or S, the game shifts to the oval in the upper middle, as indicated by the dashed arrow, whose norm calls on Row to “apologize.” (An assumption behind the rules for Row’s moving is that Column always follows the appropriate norms.)
Turning to the support graph, if Row’s discounted present value at “Row Apologizes” is lower than at “Row Cooperates,” the prospect of staying at cooperation induces Row to follow the rule at that state, while the (worse) prospect of entering the apology state deters Row from doing an alternative. Thus the norm at the apology state thus supports the norm at the cooperation state. The incentives at every state generate behavior consistent with the equilibrium, by either inducing Row to stay there or pass on to somewhere else. The inducement and deterrent aspects of the norm associated with each state correspond to social rewards and punishments, the defining aspects of norms.
The whole regime comprises five norms – one for what you should do in normal play, two for the sequence of what should do if the other commits a violation of any norm, and two for the sequence of what you should do if you yourself commit a violation of any norm. Note that if Row apologizes (inappropriately) during mutual cooperation (i.e., moves to the bottom left oval), then play goes to Row’s punishment path  Row must apologize and restitute for inappropriately apologizing. Also, Row’s failing to “accept” Column’s restitution (by nonnormatively playing Transfer at the top right box) calls for an apology sequence from Row. So does Row transferring goods to Column when Row should be letting Column apologize. Apologizing in these situations seems odd and in reality someone who constantly apologized for nothing might be sanctioned, but that would happen in other ways. The network is set up this way to minimize the number of norms.
It is interesting why the ApologizeandRestitute equilibrium is the most robust in its category, but the point of the example is to illustrate what a normative system around apologies might look like, and to show it can be given a strategic basis. Apologizing is a general purpose supporting norm since violations of other norms usually require an apology, and indeed one can argue that failing to apologize itself merits an apology. That apologies support themselves is the crucial element in keeping the set of norms small. An apology for not apologizing is usually left understood, to keep the interaction smooth and simple. Still, when President Clinton apologized for the 1930s governmentsponsored syphilis experiments in Tuskegee, Alabama, he included this further element (1997), “The American people are sorry  for the loss, for the years of hurt. You did nothing wrong, but you were grievously wronged. I apologize and I am sorry that this apology has been so long in coming.”
An unusual aspect of apologies as supporting norms is that they call for the wrongdoer to participate in his own punishment. The stereotypical view of social norms is that the violator is punished by others, but here the others act only when he or she refuses to do it. One can imagine psychological reasons for this in that the apologizer then “owns” the wrong and may feel less resentment than if others were inflicting the harm.
Contrarytoduty obligations in deontic logic and game theory
The strategic approach can be compared with deontic logic, the formal method most commonly used in the philosophy of ethics. Deontic logic is concerned with the logical relationships among statements about obligations of agents or ideal states, and is nonstrategic, without either probabilities or utilities.
A wide debate in that literature is on contrarytoduty obligation, which arise when the party has violated some duty and thereby incurred a new one (Chisholm 1964). I am obliged not to commit a murder, one example goes, but if I do that anyway I should do it gently. The latter is a contrarytoduty obligation, but it seems odd to proclaim an ethical rule for how to do something wrong. Apologizing is a contrarytoduty obligation – I should not insult someone but if I do, I should apologize for it. A “paradox” of such obligations, adapted here from a version of Forrester (1984), shows a technical problem for a deontic logic analysis. Let the proposition i mean that I insult someone, let a mean that I apologize for the insult, and let Oblig a mean that I am obliged to apologize. We assume these premises:

Oblig ~i

(1)

i Oblig a

(2)

a i

(3)

i

(4)

The first is selfexplanatory, (2) is the contrarytoduty obligation, while (3) states that I cannot apologize for something that I did not do. I may say the words, “I am sorry for having started the Hundred Years War,” but this is a pretense, and cannot be a real apology. Premise (4) states that I did indeed insult the person so am facing my contrarytoduty obligation. As well as the inference rule of modus ponens, the following is included, a standard one in deontic logic:
p q → Oblig p Oblig q (closure).
For example, if going to work means getting out of bed, it follows that if I am obliged to go to work then I am obliged to get out of bed.
From (2) and (4),
Oblig a (5),
and from (3) and closure we have,
Oblig a Oblig i (6).
Combining (5) and (6) with modus ponens,
Oblig i (7).
So together (1) and (7) require me to insult and to refrain from insulting the person – a contradiction.
The difficulty seems to be that the closure rule is too broad, but just how to restrict it continues to be debated, with proposals that are either flawed or extremely complicated. From a game theory viewpoint the problem is that standard deontic logic has no rule account of the connection between the original obligation and the contrarytoduty one. Writers sometimes call the latter a “secondary obligation,” as if I am bound by (1) but if I somehow violate it I should try for (2) as second best. This ignores the strategic issue. To represent (2) as the incentive for following (1), one needs to incorporate goals and beliefs. The notion of behavior offtheequilibriumpath is familiar to game theorists, as well as the idea that what would hypothetically happen there determines real behavior. This is not to downplay attempts to solve this problem within deontic logic, only to point out that the strategic approach easily captures one feature that the logical one misses.
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