The Predictioneer’s Game: Using the Logic of Brazen Self-Interest to See and Shape the Future

Appendix to the Paperback Edition

Because some of you asked for it:

The figure on this page illustrates a single stage game for a single pair of players (call them A and B) while not displaying the sources of uncertainty in the model. Nature assigns initial probabilities of 0.5 to player types, and the model then applies Bayes’ Rule so that the players can update their beliefs. There are sixteen possible combinations of beliefs about the mix of player types. Each player is uncertain whether the other player is a hawk or a dove or whether the other player is pacific or retaliatory. By hawk I mean someone who prefers to compel a rival to give in to the hawk’s demands even if this necessitates both imposing and enduring costs rather than compromising on the policy outcome. A dove prefers to compromise rather than engage in costly coercion to get the rival to give in. A retaliatory player prefers to defend himself—at potentially high costs—rather than be bullied into giving in, while a pacific player prefers to give in to avoid further costs.

The game is iterated so that payoffs can change from round to round, with a round defined as a sequence of moves through the stage game in the figure. Because the game is solved for all directional pairs, it assumes that players do not know whether they will be moving first, second, or simultaneously with each other player. The game ends, by assumption, when the sum of player payoffs in an iteration is greater than the projected sum of those payoffs in the next iteration.

Structure of the Game: Sketch of One of N²–N Stage

Games Played Simultaneously

A and B uncertain whether other will retaliate if coerced

A and B uncertain whether other will coerce if given the chance

In playing the game, each player’s initial move is to choose whether to make proposals. A proposal is expressed as a demand that another player accept some specific change in his position that is favorable to the demander. Players choose proposals designed to maximize their welfare at the end of the stage game. In practice, this means choosing proposals that make the other players indifferent between imposing costs on the demander and preferring a negotiated compromise instead. Of course, the endogenous selection of proposal values must take into account player beliefs about their rival’s type.

Payoffs are calculated as follows:

Let the probability that A prevails in an iteration of the game vs.

C is the potential clout or influence of each stakeholder, S is the salience each stakeholder attaches to the issue, and U denotes utility with the first subscript indicating whose utility is being evaluated and the second vis-à-vis which other player’s approach to the issue.

Let X1_k = player K’s policy preference on the issue; Let X2_k = player K’s preference over reaching agreement or being resolute on the issue.

Let A’s utility for B’s approach to the issue with that is, the model assumes that players prefer a mix of gains based on sharing resolve or flexibility to settle and based on the issue outcome sought over fully satisfying themselves on one dimension while getting nothing on the other. The structure of the utility of proposals is comparably computed but with positions chosen endogenously rather than necessarily being either player’s policy position.

The model assumes four sources of costs:

(1) α, the cost of trying to coerce and meeting resistance; (2) τ, the cost of being coerced and resisting; (3) γ, the cost of being coerced and not resisting; and (4) , the cost of coercing; that is, the cost of failing to make a credible threat that leads the foe to acquiesce. It also allows all of the input variables to change (doing so in accordance with heuristic rules I impose on the game). That is, the model is designed so that player clout, salience, resolve, and position shift from iteration to iteration in response to the equilibrium conditions of the prior round of play. Because alternative heuristic rules chosen by others are likely to be as sensible and reliable as mine either for evaluating costs or for assessing how variable values change across periods of play, I do not dwell here on those aspects of the model.

With these values in hand, here are the expected payoffs at terminal nodes in the first iteration:

D* and R* denote, respectively, the belief that the subscripted player is a dove and a retaliator. These beliefs are updated in accordance with Bayes’ Rule. Off-the-equilibrium path beliefs are set at 0.5.

Proposals go back and forth between players, but not all proposals are credible. They are credible if the Outcome involves B giving in to A’s coercion or if the absolute value of the proposal being made minus the target’s current position relative to the range of available policy differences is less than the current resolve score of the target, with resolve defined in the next section.

The predicted new position of each player in a given round is determined as the weighted mean of the credible proposals it receives and the predicted outcome is the weighted mean of all credible proposals in the round, smoothed as the average of the weighted means including the adjacent rounds just before and after the round in question. The nature of the proposal in each dyadic game is determined by the equilibrium outcome expected in that stage of the game. The weighted mean reflects the credibly proposed positions weighted by clout multiplied by salience.

Phew, I hope the math mavens enjoyed that and the rest of you didn’t mind it too much.