Market Entry and Game Theory: Incumbent vs. New Entrant
Market Entry and Strategic Interaction: A Game Theory Perspective
**a) Payoffs with c=0**
There are three possible outcomes of the game, depending on the choices of the new entrant (entry vs. no entry) and of the incumbent firm (aggressive vs. passive behavior). In case of no entry, the incumbent will set the monopolist level of output. Let i be the incumbent label and e the new firm label. The monopolist quantity is qmi = 1/2 and the payoffs are Πmi = 1/4 and Πme = 0.
If the new firm enters the market and the incumbent plays passively, Cournot competition arises with qci = qce = 1/3 and Πci = 1/9, while Πce = 1/9 − 1/10 = 1/90. The two firms produce the same quantity but earn different profits, given the fixed cost the new firm has to pay to enter the market.
Finally, if the new firm enters the market and the incumbent behaves aggressively (i.e., it produces the quantity such that p = c), then the new entrant pays the fixed cost but is not able to produce a positive quantity since firm i serves all the market at p = 0. So, qai = 1, qae = 0, Πai = 0, and Πae = −1/10.
**b) Game Tree**
- E = I2
- Passive {1/90, 1/9}
- Aggressive {−1/10, 0}
- I1 Do nothing {0, 1/4}
**c) Is Aggressive Behavior Credible?**
To assess the credibility of aggressive behavior, we rely on the notion of Subgame Perfect Nash Equilibrium (SPNE). This is the Nash equilibrium of the dynamic game in which the equilibrium strategies of the players are optimal in each subgame since in each subgame, the players are maximizing their payoff. To find the SPNE, we use the backward induction technique by analyzing the last subgame, solving for the best choices, and going backward.
Starting from the node I1 (that is, we are assuming that in the first stage the new firm enters the market), the incumbent firm strictly prefers to be passive rather than aggressive; in fact, the payoff for the incumbent is 1/9 in the first case and 0 in the latter. At node I2, there is no choice, and firm i simply behaves as a monopolist, given the no-entry decision of the new firm.
Given these solutions to the last stage of the game, in the first stage, the new entrant has to choose whether to enter the market. If it does not enter, its payoff is zero, while by entering, it gets its Cournot profits. The new entrant will choose to enter because it anticipates that the incumbent will be passive because the threat of aggressive behavior is not credible (i.e., it is not in the interest of the incumbent when it has to make the choice).
**d) Table**
To use the normal form to find the Nash equilibria of the dynamic game, it is necessary to specify the strategies of the two firms. Recall that in a dynamic game, a strategy for a player is a set of instructions identifying what decision to take at each relevant node. So, player e has two strategies: enter and not enter. Firm i has two strategies: {aggressive, do nothing} and {passive, do nothing}.
The normal form representation of the dynamic game is:
| Aggressive, Do Nothing | Passive, Do Nothing | |
|---|---|---|
| Enter | −1/10, 0 | 1/90, 1/9 |
| Stay Out | 0, 1/4 | 0, 1/4 |
There are two Nash equilibria: (stay out, {aggressive, do nothing}) and (enter, {passive, do nothing}). The first one is not credible for the reason stated in the previous point.
**e) SPNE**
The unique SPNE of this repeated game consists of the repetition in each of the 10 stages of the game of the SPNE of the one-shot sequential game described in the solution to the previous point. This is because the game is finite, and in the last stage, the incumbent has no reason not to accommodate entry because there will be no further stage, and the payoff of behaving passively is larger than the payoff of behaving aggressively.
At stage 9, the only reason to behave aggressively is to threaten the new firm in stage 10. But knowing that entry will be accommodated anyway, this is not a useful (and so credible) strategy for the incumbent. Then, the incumbent will be passive in stage 9, too. The same reasoning may be applied to all the previous stages to get the SPNE of the full repeated game.
Generalized Second-Price Auction Analysis
**a) Nash Equilibrium**
If j ≤ i, the inequality is obviously true. Otherwise, consider the player π(j) in slot j. Since b is a Nash equilibrium, the player in slot j is happy with her outcome and does not want to increase her bid to take slot i, so:
αj(γπ(j)vπ(j) − γπ(j+1)bπ(j+1)) ≥ αi(γπ(j)vπ(j) − γπ(i)bπ(i))
Since bπ(j+1) ≥ 0 and bπ(i) ≤ vπ(i), then:
αjγπ(j)vπ(j) ≥ αi(γπ(j)vπ(j) − γπ(i)vπ(i)).
**b) Price of Anarchy**
Taking j = σ(i) in the definition of weakly feasible allocations, we get that:
ασ(i)γivi + αiγπ(i)vπ(i) ≥ αiγivi.
Now, summing this for each player i, we get:
2 · SW(π(b), v) = Σi ασ(i)γivi + Σi αiγπ(i)vπ(i) ≥ Σi αiγivi = OPT(v).
**c) Bound on Price of Anarchy**
For two slots, consider an example with two players with valuations 1 and 1/2, respectively, quality factors γ1 = γ2 = 1, and two slots with α1 = 1 and α2 = 1/2. The bids b1 = 0 and b2 = 1/2 are at equilibrium, resulting in a social welfare of 1, while the optimal social welfare is 1.25.