Math 2.0

Game theory, economics, finance, and other social and behavioral sciences Definition

Differential game

Builds on Control systemSaddle pointLipschitz continuity

Definition

Let N≥2N \ge 2 and T>0T > 0. For each player i=1,…,Ni = 1, \dots, N, let Ui⊆ℝmiU_i \subseteq \R^{m_i} be a nonempty compact set. Let f ⁣:[0,T]×ℝn×U1×⋯×UN→ℝnf \colon [0, T] \times \R^n \times U_1 \times \cdots \times U_N \to \R^n be continuous and Lipschitz in its second variable. For each ii, let Li ⁣:[0,T]×ℝn×U1×⋯×UN→ℝL_i \colon [0, T] \times \R^n \times U_1 \times \cdots \times U_N \to \R and gi ⁣:ℝn→ℝg_i \colon \R^n \to \R be continuous. A differential game on [0,T][0, T] with initial state x0∈ℝnx_0 \in \R^n consists of these data, used as follows.

  1. Each player ii chooses a measurable control ui ⁣:[0,T]→Uiu_i \colon [0, T] \to U_i.
  2. The controls drive the state by the control system x′(t)=f(t,x(t),u1(t),…,uN(t))x'(t) = f(t, x(t), u_1(t), \dots, u_N(t)) with x(0)=x0x(0) = x_0.
  3. Player ii wants to minimise its cost Ji(u1,…,uN)=∫0TLi(t,x(t),u1(t),…,uN(t)) dt+gi(x(T))J_i(u_1, \dots, u_N) = \int_0^T L_i(t, x(t), u_1(t), \dots, u_N(t)) \, dt + g_i(x(T)).

The game is two-player zero-sum if N=2N = 2 and J2=−J1J_2 = -J_1; then player 11 minimises J=J1J = J_1 and player 22 maximises it.

Remarks

A full definition also says what information each player uses. In open-loop play each player picks a control as a function of time alone. In feedback play each player picks ui(t)u_i(t) as a function of (t,x(t))(t, x(t)). In the zero-sum case one often uses nonanticipating strategies, which choose the present control from the opponent's past controls. A solution concept is a Nash equilibrium, a tuple of strategies from which no player gains by deviating alone. In the zero-sum case this is a saddle point of JJ, and its value is the value of the game.

Example: n=2n = 2, U1U_1 the closed disc of radius 22, U2U_2 the closed unit disc, f(t,x,u1,u2)=u2−u1f(t, x, u_1, u_2) = u_2 - u_1, L1=0L_1 = 0 and g1(x)=∣x∣g_1(x) = \abs{x}, with g2=−g1g_2 = -g_1. Here xx is the position of player 22 relative to player 11, and player 11 wants to end close to player 22. All clauses hold: ff is continuous and does not depend on xx, and the sets UiU_i are compact.

Non-example: in a game where the state changes only at the times 0,1,2,…0, 1, 2, \dots by a recurrence xk+1=F(xk,u1,k,u2,k)x_{k+1} = F(x_k, u_{1,k}, u_{2,k}), clause 2 fails. It is a dynamic game in discrete time, not a differential game.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Control system, concept, no evidence levelControl systemconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptLipschitz continuity, definition, no evidence levelLipschitz continuitydefinitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptLocal maximum and minimum, concept, no evidence levelLocal maximum and min…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptGradient, concept, no evidence levelGradientconceptSaddle point, concept, no evidence levelSaddle pointconceptDifferential game, definition, no evidence levelDifferential gamedefinition
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  1. Control system concept
  2. Euclidean space object
  3. Function concept
  4. Lipschitz continuity definition
  5. Total derivative concept
  6. Function of several variables concept
  7. Differentiable function of several variables concept
  8. Local maximum and minimum concept
  9. Partial derivative concept
  10. Gradient concept
  11. Saddle point concept
  12. Differential game definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Differential_game
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URIm2://TJS0MZSFA8
Kinddefinition
WritingW0 stub
MSC91A23, 49N70
DefinesDifferential game
UsesControl system, Saddle point, Lipschitz continuity

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