A differential game studies system dynamics determined by the interactions of agents with divergent purposes. As a limit form of multi-stage games, its non-cooperative solution is subgame perfect; thus it may facilitate the study of credible threats and repeated play. Reducing each stage to a single point in continuous time, differential game applies control theoretic tools (including phase diagrams) to yield results more general and more detailed than other methods. Its applications range from common-property resource utilization to macro-economic stabilization.

Model

A differential game has four components: (a) a state space, X, where x in X embodies all relevant data at a particular stage, (b) a time horizon, T: a closed interval with a final instant equal to infinity or decided by some termination rule, (c) a set of players,\( \overline{N}=\left\{1,\dots, i,\dots, N\right\}, \) with each player distinguished by four aspects: (1) a space for possible moves (or ‘controls’), Ki; (2) a point-to-set correspondence for allowable moves, Ci: X × T ⇒ Ki, (x, t) ↦ Ci(x, t) which vary with (x, t); (3) a space for admissible ‘strategies’ (or ‘policies’), Ri = {riri: X × T → U X × TCi(x, t) and ri satisfies additional conditions}, where each ri assigns an allowable move at every (x, t); and (4) the instantaneous payoff ui: X × T × ∏jKj → R, ((x, t), (cj)) ↦ ui((x, t), (cj)). The additional conditions include (i) any restrictions on the information used for decisions, (ii) regularity conditions (e.g., being step-wise continuous) needed for a well defined model (d) a state equation for state transition, \( F:X\times T\times {\varPi}_j\;{K}_j\to X,\left(\left(x,t\right),\left({c}_j\right)\right)\mapsto \overset{.}{x},X \) and K1 ,…, Kn are all subsets of Euclidean spaces.

Players select strategies at the outset, not piecemeal moves. Strategies are defined here as state-and-time dependent or ‘feedback’ strategies including the subclass which are ‘open-loop’ or time-dependent (only).

An Example with Two Variations

Two users share a natural resource, which may be a petroleum reserve or fishery, under common-property tenure. The state space X is the set of all non-negative resource levels and the time horizon is T = [0, tf] where tf = +∞ or the instant when all the resource is used up. For all (x, t), i = 1, 2, the ‘allowable moves’ form a set Ci(x, t) = Ki = R+, the set of all non-negative rates of use. Specimens of strategies include ri = kx for some k ≥ 0, or ri = g(t) for some non-negative-valued function. The instantaneous payoffs of both players are assumed to be: ui = exp(−at) log ci for some a > 0. The ‘state equation’ is: dx/dt = f(x) ∝ − c1 − c2 where: f(x) = 0 for the case of petroleum reserves, and f(x) = x(b − log x) for the fishery. The latter form agrees with the Gompertz recruitment function.

Solution Concepts

How players choose strategies under various scenarios is summarized as three solution concepts, e.g. (1) The noncooperative equilibrium (Cournot–Nash): each player’s choice is his ‘best reply’ to the choices of all other players. This choice must be ‘best’ for all initial (x, t); (2) The cooperative equilibrium (Pareto): all players make choices such that no modification can benefit any player without harming another. This property holds for all initial (x, t); (3) The hierarchical equilibrium (Stackelberg): the ‘leader’ selects a committed choice to elicit the followers’ ‘best replies’ so that the leader’s payoff is maximized.

An equilibrium is a vector of strategies, one for each player, which is not liable to change. In differential games, players may change strategies in midgame, unless prevented by prior commitment (as in (3)), or by requiring the choices to be appropriate, once and forever (as in (1) and (2)). Significantly, the Cournot–Nash solution is thus subgame perfect à la Selten.

The Cournot–Nash solution is most frequently used. In particular, it depicts externalities under laissez-faire. For any problem it can be compared to Pareto solutions which assess any extra gains resulting from cooperation.

If an acknowledged leader (e.g. the government in a macroeconomy) can offer credible commitments, he prefers to play Stackelberg (with a higher payoff for himself) rather than Cournot–Nash, since all followers’ best replies are now under his influence rather than given independently.

The differential game sheds light on two additional features: (i) the credibility of the leader’s professed strategy which is at issue, since he has both (a) the opportunity to renege on promises made and honoured at different times, and (b) the incentive to renege (his choice is subgame imperfect); (ii) ‘Reputation’ (rather than ‘enforcement’) is often the reason why commitments are kept, and may be modelled as a state variable suggested in Clemhout and Wan (1979). Hence credibility is established from a balance of the gains of reneging with the damage from reputation lost. This suggests a synergistic approach between Cournot–Nash and Stackelberg.

Alternative Formulations of Cournot–Nash Models Over Time

To characterize the Cournot–Nash differential game in feedback strategy, one contrasts it with alternative versions of differing assumptions, ‘what information players use’ and ‘the modelling of time’. Examples show that:

  1. (a)

    To explain reality and provide policy relevance, ‘feedback’ strategies are preferable to ‘open-loop’ strategies for two reasons; (1) in non-cooperative games, the subgame-perfect equilibrium is the image of reality, and (2) in models of common-property resources, policy relevance hinges on identifying the source of inefficiency. Our petroleum example (cf. Clemhout and Wan 1985a) is a non-cooperative model of common-property utilization, and thus should have an inefficient but subgame-perfect equilibrium. This is the case when strategies are ‘feedbacks’. The opposite is true if strategies are modelled as ‘openloop’ in which all equilibria are then efficient and subgame-imperfect and each is compatible only with one initial resource stock.

  2. (b)

    For reasonableness and convenience, ‘history-dependent state variables’ are preferable to ‘history-dependent strategies’. While history matters in contexts such as performance contracts, history-dependent strategies tend to require an infinite amount of information at every move. The use of history-dependent state variables (Smale 1980, cf. Clemhout and Wan 1979) is a reasonable alternative for players with bounded rationality. It also conforms to the finite-dimensional state space in differential games.

  3. (c)

    For game-theoretic and analytic reasons ‘continuous time’ is preferable to ‘discrete time’. Our fishery example (Clemhout and Wan 1985a) illustrates two points. First only in continuous time is the model a game, according to Ichiishi (1983). To ensure non-negativity of the resource level, discrete-time models require the allowability of one player’s move to depend upon the moves of all others at the same time, thus they become ‘pseudo-games’ by losing playability. The second point is that only in continuous time are the dual variables (which are analytically important) derivable from the conditions necessary for optimality whether the recruitment function is concave or not. This is because the adjoint system, in differential equation form, involves the slope of the recruitment function alone and not its curvature.

Strengths of Differential Games

Differential games can obtain precise results either independently of particular functional forms, or by using empirically validated formulations. In our fishery example these include characterization of the resource level: (a) does it reach a sustained level? (b) does it approach extinction asymptotically, if so, how rapidly? (c) is it heading for extinction in finite time? (d) what difference do risks of random perturbation or extinction make? (e) what if several harvested species form pre-predator chains? and (f) do tax-incentives improve allocation efficiency? (Clemhout and Wan 1985a, b, c).

In contrast with differential games, intuitive reasoning or simple examples (in two or three periods) can suggest certain outcomes, but cannot rule out the opposite outcome occurring in plausible situations. Simulation models can start from any assumptions but cannot assure equilibrium.

In macro-economics, the linear-quadratic-Gaussian differential game can further analyse quantitatively real-life data. The estimation and interpretation of the parameters in such models is still subject to ongoing research. The same model also yields deep economic insights in their micro-economic applications.

Concluding Remarks

Pioneered by Isaacs and generalized by Case, the theory of differential games is now covered by excellent texts (e.g., Basar and Olsder 1982), with reference to contributions by Blaquiere, Berkovitz, Cruz, Fleming, Friedman, Haurie, Ho and Leitmann, among others. Further progress in its economic applications now hinges on the development of ‘techniques of analysis’, akin to phase diagrams in control theory. Using these techniques one can deduce implications crucial to economists working with particular classes of models. This is often accomplished by utilizing structural properties common to entire families of models. The explicit solutions are neither required nor derived. Such feats are clearly attainable for differential games, as they have been for control models: the phase diagram itself has been recently applied to some models (Clemhout and Wan 1985b) and contraction mappings in others (Stokey 1985). Given the state of the art in this field, additional advances in theory (e.g., generalizing the model, proposing new solution concepts, etc.) are certainly most welcome, but no longer crucial for economic applications.

See Also