Figure 3.

The complexity of strategies and social norms. Under IR, and even in a world of binary reputations and decisions, social norms and strategies can rely on arbitrarily large information sets and encapsulate arbitrarily complex decision-making rules. The level of information used can be captured by the order of social norms, which translates the number of information layers that a norm can use: 0th-order norms prescribe unconditional judgements, such as everyone is good (all good); 1st-order norms discriminate based on the donor's action—such as the well-known IS norm that assigns G only when donors cooperate; 2nd-order norms include information about the recipient's reputation; 3rd-order norms include information about the donor's reputation; and finally, 4th-order norms include information about the previous reputation of the recipient [32]. Naturally, other sources of information could be considered (e.g. [81]), possibly expanding a norms' order beyond the 4th. Order is represented by the rows in the panel above. Within each set of norms with a given order, information can be combined to formulate judgements with a variable degree of complexity (columns). Relying on the binary value of reputations and actions, we capture complexity through the so-called Boolean complexity [32,48]. In the figure, we provide a visualization according to the layout of strategy and social norm tables represented in figure 2. Following the method of Karnaugh maps [82], the complexity of norms and strategies can be determined by counting the number of blocks of size 2^k ‘G's or ‘Cs’, until all coloured cells are covered and where k is chosen to be as large as possible and blocks can overlap: each block of 2^k ‘G's or ‘Cs’ increases the complexity of norms by 4 − k and strategies by 3 − k. (Online version in colour.)