2nd- Dynamic Games

  In the remainder of this chapter we will briefly discuss dynamic games, and in particular the difference between static and dynamic games. Dynamic games are, put simply, games with a time aspect in them. For example, if one firm acts before the other, this has quite important implications for playing the game: the second firm can play the game knowing what the first firm has done, whereas the first firm has to make its decision without the requisite knowledge about the follower. Some games simply don’t make much sense to play sequentially – paper/scissor/ stones, for example, would not be very exciting if one player knew what the other player has chosen.3 Some games, on the other hand, could be played either simultaneously or sequentially. Setting prices, for example, will be done without knowledge of rivals’ prices some of the time (making it a simultaneous game), but in other situations sequential moves might be more relevant. Representing a sequential game is usually done by drawing a game tree, where the first decision starts the game, and every decision point represents a node from which the decisions of subsequent players branch out accordingly. We illustrate this with an example. Assume that two firms selling mineral water have to decide on their advertising budget. For simplicity, there are only two levels of advertising, High and Low. Harrogate Spa chooses its advertising level first, and Vittel chooses after that. If both firms choose H, profits are zero (because all the money is spent on advertising), if both choose L, their profits are £1m each, and if one chooses H and the other L, the firm running an intensive advertising campaign makes £1,250k, while the other one makes profits of £500k. A game tree (or extensive form) of this situation would then look like in the figure.

 Game tree.

How should one analyse a sequential game? Analysing sequential games has a very similar objective to analysing simultaneous ones: predict sensible behaviour and an eventual outcome of the game. The way to do this in a simultaneous game is by eliminating dominated strategies and/or playing dominant strategies. (While this may enable us to simplify the game, it may still not lead us to a Nash equilibrium. In this case we would have to look for Nash equilibria after having simplified the game as much as we could.) One way of simplifying a sequential game is by backward induction. Backward induction works by eliminating strategies at the final node of the game (i.e. the point when the last player makes a decision, based on the decisions previously taken), and working one’s way forward, that is, closer to the start of the game. The strategies we can eliminate are moves that would not maximise an individual’s profit at that point. A rational player should never select these moves, which means that a player anticipating rivals’ moves should not expect these moves to be chosen. Looking at the game referred to above we can see that Vittel would not choose H if Harrogate Spa plays H, and they would not choose L if Harrogate Spa chose L. This then enables Harrogate Spa to anticipate that playing H gives profits of £1,250k, while playing L yields profits of £500k – Harrogate Spa should then choose high advertising expenses, which Vittel will react to by choosing low expenses.

Commitment 

We already mentioned commitment in the context of selecting among different possible equilibria in a static game. Commitment, however, can also be used by an agent to choose the preferred course of action that he would not otherwise choose. Again, we use an example to illustrate this point. Suppose a monopolist (M) in a market faces a potential entrant (E). The entrant can choose whether to enter (e) or not (ne). If E does enter, the monopolist can choose to fight (f) or to accommodate (a). If E does not enter, M experiences business as usual. Suppose M currently makes profits of PM = 50 (and E makes zero profits in this market), in the case of entry and accommodation the entrant makes PE = 10, the incumbent PM = 20, and if entry is followed up by fight (think of this as the monopolist starting a price war flooding the market), profits are PM = PE = – 10.4 

 

By backward induction we now find that the Nash equilibrium is for E to enter and for M to accommodate – if entry does occur, M will choose payoffs of 20 over –10, and, knowing this, will prefer entering to not entering. The monopolist will not be particularly happy with this outcome: of course, given the choice, he would rather keep the entrant out – for example, by committing to fight in the case of entry. To this end, M could issue a statement along the following lines: This seems sensible enough, and should go some way towards convincing the entrant not to enter. Or should it? The entrant would have to believe that M would indeed prefer to fight. This would imply, however, that in the case of entry M would choose a sub-optimal action, namely one that gives him pay-offs of –10 rather than 20. In game-theoretic terminology, fighting after entry is not sub-game perfect – a sub-game starts at one player’s decision node and covers all the decisions that follow on from this node, and an action that does not maximise pay-offs at that decision node should not be played. Since post-entry fighting is not sub-game perfect, the entrant should not believe the monopolist if he makes this statement. Let us now consider another strategy by M. Suppose that M sign and publicise a long-term contract with one of suppliers that states: ‘if we, M, ever purchase less than the current quantity, we will incur contractual penalties of 40.’5 This seems like an odd move to begin with, since all that it achieves is to lower pay-offs in some cases, but it never increases pay-offs. However, it changes things around in the particular game we are analysing. Accommodating implies sharing the market, that is, selling less and consequently buying less from one’s supplier. This means that it is now less attractive to accommodate, since M would have to incur contractual penalties, taking down profits from accommodating from +20 to –20. This changes the pay-offs and the way the game is being played – it is now sub-game perfect to fight after entry, which means that the entrant has to choose entering and incurring losses or keeping out and have unchanged profits. The entrant will choose to stay out, leaving the incumbent with profits of PM = 50. What happened? By limiting his options, the monopolist was able to commit to playing the game differently, which accordingly made the entrant play the game differently as well, taking M’s expected reaction into account. If we represent this in our game tree 

, the monopolist changed the pay-offs in one game outcome and subsequently the sub-game perfect equilibrium.

In this game, committing credibly was worth 30 – the difference between the outcome of the game with commitment and the outcome without. Committing can be done by limiting one’s options, or lowering pay-offs in some (undesired) states of the game.