Game Applications in Finance and Economics

Economics as a Field for Games

Financial markets, oligopolistic competition, climate negotiations—all of these are tasks of strategic interaction in a dynamic environment. Differential games provide a rigorous mathematical apparatus for their analysis. The key advantage over static game theories is the consideration of dynamics: capital accumulation, changes in market share, resource depletion. Without dynamics, many important effects (for example, why countries “still” do not abide by climate agreements) simply are invisible.

Cournot Duopoly in Continuous Time

Model: Two producers release $q_i(t)$ units of goods. Price depends on total output: $P(t) = a − b(q_1(t) + q_2(t))$.

Capital dynamics: $\dot{K}_i = u_i − δK_i$ (investment $u_i$, depreciation $δ$). $q_i = αK_i$.

Functional: $J_i = \int_0^\infty e^{−ρt} [P(t)q_i(t) − c u_i(t)] dt$.

Nash equilibrium: via Hamiltonian equations or DP. Optimal investments $u^*_i(t)$ depend on own capital and the competitor’s capital.

Stationary Nash state (as $t → ∞$): Each producer achieves a stable level of capital $\bar{K}_i^* = \arg\max$ profit at equilibrium prices.

Comparison with cooperative: Cooperative players produce less (monopoly output), earn more in total. Nash—higher total output, lower price, “social losses” are smaller. Prisoner's dilemma in dynamics!

Arms Race: Richardson Model and Its Game Generalization

Richardson Model (1919): Dynamics of armaments of two countries:

$\dot{x} = a y − m x + g_1$ (country 1), $\dot{y} = b x − n y + g_2$ (country 2)

$a, b$ — "response" to the opponent’s armaments, $m, n$ — "economic deterrence", $g_1, g_2$ — "grievances".

Equilibrium: $\dot{x} = \dot{y} = 0 → x^* = (an + g_2b + g_1n)/(mn − ab)$, similarly $y^*$.

Stability: $mn > ab$ → stable equilibrium. $mn < ab$ → unstable arms race (armaments grow until war or disarmament). This is the classic model of the Cold War!

Game generalization: Each country optimally chooses $u_i$ (rate of arms increase). $J_i = \int e^{−ρt} [−c u_i^2 + π_i(x, y)] dt$. Nash equilibrium via Riccati equations—gives the "optimal" arms race.

Conclusion: Nash equilibrium of the "race" is inefficient—total arms expenditures in NE are higher than with a cooperative agreement. This is a mathematical justification for why arms control is beneficial to both sides.

Competition in R&D (Spence Model)

Model: Two firms invest in reducing costs. Accumulated R&D $k_i$ reduces marginal costs: $c_i(k_i) = c_0 − αk_i$. Each firm: $\max J_i = \int_0^\infty e^{−ρt} [π_i(c_1,c_2) − u_i] dt$ with $\dot{k}_i = u_i$.

Cournot profit: $π_i = \frac{(a − 2c_i + c_j)^2}{9b}$.

Nash equilibrium: via Riccati equations. Both firms invest, competing to reduce costs. "Innovation race".

Effect on society: Nash investments in R&D vs optimal? With spillovers ($α > 0$—one firm "learns" from another): Nash investments may be lower or higher than socially optimal. Patent system influences this balance.

Mean Field Game in Finance: Optimal Liquidation

Task: $N$ traders want to sell their positions $x_i(0)$ within time $T$. Sales affect price: $P(t) = P_0 − λ \sum_i \dot{q}_i(t)$ (aggregate “price impact”).

With $N → ∞$: Each trader optimizes against the mean field $Q(t) = E[\dot{q}(t)]$.

MFG equations: HJB: $−\frac{\partial V}{\partial t} + λQ \frac{\partial V}{\partial x} + \frac{(\frac{\partial V}{\partial x})^2}{2λ} = 0$. FPK: $\frac{\partial m}{\partial t} + \frac{\partial[m \cdot H_pV]}{\partial x} = 0$.

Solution (explicit for linear price impact): All traders liquidate at speed $\dot{q}^*(t) = x_0 \cdot \frac{T−t}{T^2}$—"uniform liquidation". This is optimal strategy under weak interaction.

With strong interaction (large $λ$): Traders "synchronize" and sell simultaneously → flash crash! MFG mathematically explains how rational individual behavior leads to systemic risks.

International Environmental Agreements

Model: $N$ countries, total emissions $E(t) = \sum_i e_i(t)$. CO₂ stock: $\dot{S} = E − αS$ (absorption). Damage: $d_i = d_i(S)$. Benefit: $b_i(e_i)$ (profit from emissions).

Nash without agreement: Each country maximizes $\int e^{−ρt}[b_i(e_i) − d_i(S)] dt$. The external effect of emissions (damage to others) is not considered → "tragedy of the atmosphere". Emissions are higher than optimum.

Cooperative agreement: Maximize $\sum_i J_i$ → reduce emissions to the social optimum. The "gain" from cooperation needs to be distributed through transfer payments.

Stability: Small coalition will "leave" if the benefit from not complying > penalty. Dynamically coordinated distribution (Yeung-Petrosyan) ensures stability. This is the mathematical foundation of the Paris climate agreement debates.

Games in Financial Markets

Financial markets are a natural medium for differential games. Each trader tries to maximize their own profit; their actions affect prices seen by other traders. Classic settings:

Optimal execution: Sell a large block of shares in time $T$, minimizing costs and market impact. Basic Almgren–Chriss model (2001): Minimizing expected costs plus penalty for variance. Extensions for trader competition—a game between several executors.

Market making: Dealer quotes bid and ask prices, earns on spread, but risks accumulating inventory. Avellaneda-Stoikov (2008) frames this as a stochastic control problem; extensions with competing market makers—a differential game.

Predatory trading: One player learns of an upcoming large transaction by another and trades against them. This is a “predator-prey” game in finance.

Cooperative Games in Insurance

Insurance companies may reinsure each other, forming coalitions for distributing catastrophic risks. Profit distribution via the Shapley vector is a standard tool. The theory of cooperative differential games allows for consideration of the dynamics of capital and reserves over time.

Games in Macroeconomics

• Games between governments: Trade policy (tariffs, quotas), monetary policy—each country optimizes its outcome, influencing others
• Stackelberg games between central bank and market: The central bank sets the rate, the market reacts with expectations
• Games with common resources: Fishing, oil, water resources—classic “tragedy of the commons” problems, modeled as differential games
• Climate change negotiations: Countries choose their level of emission reduction; aggregate result is global warming

Applications in Real Trading

• HFT (High-Frequency Trading): Algorithmic trading systems use game theoretic models to forecast the actions of competitors
• Market microstructure: Order books are modeled as games between limit and market orders
• Algorithmic execution: VWAP, TWAP, Implementation Shortfall—all these algorithms can be interpreted as strategies in a differential game
• Crypto markets: Decentralized exchanges (Uniswap, dYdX) create new game theoretic problems (front-running, MEV)
• Risk management: Calculation of VaR (Value at Risk) considering strategic behavior of other market participants