10 - Dynamic games in SCM

UCL

Université catholique de Louvain Louvain School of Management / CESCM

SUPPLY CHAIN MANAGEMENT

Dynamic games in SCM

Per AGRELL [email protected]

Outline Dynamic games Biform games Folk theorem Two strategies –  Grim trigger –  Tit for Tat

Examples

Dynamic games Multi-period games –  With time-dependency Inventory games, salvage values, investments –  Without time-dependency Repeated games (Nothing links specific games, used more in reputation) Debo (1999), Taylor and Plambeck (2003), Ren et al. (2003)

Repeated games 1

2

3

…

Set of NE much larger than in static games May include NE not found in static games Trigger strategies –  May induce coordination if credible threat –  Implicit collusion

Biform games Two periods Period 1 –  Simultaneous move, non-cooperatively

Period 2 –  Cooperative game

Anupindi et al. (1999, 2001)

Concepts Sequential games

Folk theorem Given an infinite sequence of payoffs r1, r2, . . . for player i, the average reward of i is Given a discounting factor 0< b <1, the future discounted reward for player i is A payoff profile r is enforceable if r >= min max u(i){s(i),s(-i)} If r is a NE payoff vector for any game G, then r is enforceable. If r is feasible and enforceable, then r is a NE for the infinitely repeated G with average rewards.

Strategies Certain strategies may be used in repeated games to induce a specific payoff r –  Grim trigger –  Tit for Tat

Credible strategies A threat will work only if it is credible A credible strategy is not dominated in a subgame

Grim trigger Play COOP until any deviation (cheating) –  Then play NON-COOP (static NE) for rest of the game

Idea: –  Players compare discounted payoff of COOP vs NON-COOP. –  Cheating carries the cost of difference in profit for the rest of the game

Tit for Tat Play COOP until any deviation (cheating) –  Then play NON-COOP for K≥ 1 rounds –  Play COOP if counterpart played COOP last round.

Idea: –  With K =1, Tit-for-Tat is just the mirror of the counterpart’s action. –  Tit-for-Tat has shown to be a credible strategy in dynamic games –  The strategy is better than Grim for “trembling hands”, i.e. probabilistic moves

References Cachon and Netessine (2004) –  Game Theory in Supply Chain Analysis. Ch 6 in David Simchi-Levi, S. David Wu and Zuo-Jun (Max) Shen (Eds), Supply Chain Analysis in the eBusiness Era, Kluwer.

Plambeck and Taylor (2007) –  Optimal relational contracts; capacity investments

Agrell and Kasperzec (2006) –  Joint investment games, double moral hazard