The classic Stackelberg model (CSM) generally yields fractional units given its continuous divisibility assumption. We consider whether the first-mover in a duopoly game is hurt by rounding budgetary estimates of quantity as calculated from the CSM. We derive our benchmark for not rounding by considering a dynamic “game within a game” in which we relax the continuous divisibility assumption, and each player optimizes the deviation between their CSM quantity with a fractional remainder and an integer solution. Using a game-theoretical framework, we obtain a closed-form subgame perfect Nash equilibrium given integer constraints. We find that the leader is generally better off than under the CSM, in contrast to the follower who is often worse off. We show that a universal rounding strategy by the leader is suboptimal. As a verification test, we use a brute-force numerical search to confirm our theoretical solutions.
In a cooperative game on simplicial complexes with filtration (SF), players can join the grand coalition simultaneously according to a given sequence, but the simplicial value of a cooperative game on SF (SF game) does not consider player’s bargaining power in the formation of SF. Considering the influence of player’s joining order in the formation of SF, we introduce a relative marginal contribution (RMC) value in this restricted coalition and the generalized simplicial value (the GS value) is proposed as an extension of the simplicial value. Furthermore, because player’s joining order is deterministic in SF, which is different from a crisp cooperative game, we examine the effect of players’ order on their RMCs and the MCD value is characterized. The MCD value may motivate players to contribute to SF. As extensions of the simplicial value, both the GS value and the MCD value are based on RMC, while the MCD value is friendly to the players who join the cooperation early on.
This paper studies an attack–defense game with incomplete information. The game involves two players: the defender and the attacker. The defender locks objects. The attacker then tests the objects and obtains information with random errors. When attacking an unlocked object, the attacker wins the game. Sonin’s complex “fill-and-switch” strategy is reduced to a simple threshold rule: the attacker targets nodes with a negative signal when the sum of sensitivity and specificity is greater than 1, nodes with a positive signal when the sum is less than 1, and randomizes when the sum is 1. The defender allocates locks uniformly at random. Exact expressions for the attacker’s equilibrium strategy are obtained, as well as analytical expressions for the game value. The game value satisfies duality, reaches its minimum when the sum of sensitivity and specificity is 1, and converges to the node value at rate [Formula: see text]. This makes the LBT model a practical tool for security applications.
In this paper, we characterize the coalition worth function in a generalized TU coalitional game to explain the formation, size, and composition of two particular classes of sub-coalitions, defined as the maximal proper subcoalition (MPS) and stable proper sub-coalition (SPS). In the case of MPS, the coalition worth function provides sufficient incentive to each of some (but not all) players to form the coalition, not to integrate with player(s) outside the coalition, and prevents disintegration into a smaller coalition. We label this as the definition of MPS from a micro-perspective. On the other hand, SPS is defined based on the distribution of coalition worth among the members. We also define an MPS from the macro perspective, which is based on the sub-coalition (group) worth function. We show that in order to be an MPS, the coalition worth function has to be cohesive over the set of players forming the coalition and sub-additive over the set of players outside the coalition. However, this is only necessary but not sufficient for forming an MPS. Interestingly, we find that there can be TU coalitional games where there are MPS, but there is no maximal proper sub-coalition structure (MPSS). Next, we show that an MPS is always an SPS, but the converse is not true. Thus, the results of MPS automatically extend to SPS. Finally, we show that an MPSS and/or stable proper sub-coalition structure (SPSS) has higher welfare than the grand coalition.
In this paper, we develop a continuous-time equilibrium model of futures markets in which heterogeneous hedgers and speculators trade under explicit margin constraints. Agents maximize exponential utility and take prices as given, while the equilibrium futures drift is determined endogenously through market clearing. Margin frictions interact with risk-sharing motives to generate a nonlinear fixed-point problem linking individual optimality to aggregate consistency. The equilibrium is characterized by a coupled Hamilton–Jacobi–Bellman (HJB) and backward stochastic differential equation (BSDE) system, and we establish existence and uniqueness of the equilibrium drift under mild regularity conditions. A central contribution of the framework is a new geometric characterization of margin effects: margin constraints create no-trade regions in drift space, producing a frictional wedge that forces the equilibrium drift to be the projection of the frictionless Keynes–Hicks drift onto a margin-dependent interval. This yields a kinked and convex mapping from margin tightness to liquidity premia, with one-for-one amplification when the frictionless drift lies outside the wedge and flat sensitivity once it lies inside. Unlike Brunnermeier–Pedersen [[2009] Market liquidity and funding liquidity, Rev. Financ. Stud. 22(6), 2201–2238] and Gârleanu–Pedersen [[2011] Margin-based asset pricing and deviations from the law of one price, Rev. Financ. Stud. 24(6), 1980–2028], whose margin effects are linear and exogenous, our model delivers a fully dynamic, state-dependent amplification mechanism arising endogenously from equilibrium price formation. The structure provides a tractable foundation for empirical implementation using disaggregated trader positions. By using CFTC gold-futures data and CME volatility measures, we find strong empirical support for the model’s nonlinear amplification mechanism: hedging pressure alone has no predictive power; margin tightness exerts a negative baseline effect; the interaction between hedging pressure and margin tightness generates a statistically significant kink; and volatility sharply magnifies the impact of margin tightness, producing the largest and most significant effect in the data. These findings confirm the model’s core prediction that margin constraints generate nonlinear, state-dependent, and volatility-amplified liquidity premia, providing the first structural empirical evidence for a kinked margin-pressure mechanism derived from a full HJB–BSDE equilibrium.
