Do urban redevelopment incentives promote asset deterioration? A game-theoretic approach

Abstract

The elapsed time between a government’s announcement of its intention to redevelop and the launch of the new construction may often be quite lengthy. This study uses a game-theoretic framework to examine the effect of the option to redevelop on the quality of the existing housing stock during this extended pre-redevelopment period. We show that the benefits that accompany future redevelopment may lead to accelerated deterioration in the pre-redevelopment period. Moreover, we identify circumstances under which there exists a unique perfect Nash equilibrium where, in order to discourage objections by other homeowners, those who support redevelopment intentionally promote structural deterioration during the pre-redevelopment period. Our results highlight the need to shorten the period of time between the announcement of the option to redevelop and its implementation.

Appendices

Appendix 1: Proofs

Proof of Results 1 and 2

If redevelopment is exogenously imposed on the owners, then regardless of the action of Owner 1 (see nodes <2> and <3> in Fig. 1), if Owner 2 chooses I, his payoff increases by $\alpha Q^{m} - C_{2}^{m}$. Similarly, if no-redevelopment is exogenously imposed on the owners, then regardless of the action of Owner 1, if Owner 2 chooses I, his payoff increases by $2\alpha Q^{m} - C_{2}^{m}$. Hence, when redevelopment (no-redevelopment) is exogenously imposed, choosing I is Owner 2’s dominant action if and only if $C_{2}^{m} < \alpha Q^{m}$ (if and only if $C_{2}^{m} < 2\alpha Q^{m}$); otherwise, NI is Owner 2’s dominant strategy. Further, when redevelopment (no-redevelopment) is exogenously imposed, and given that Owner 2 follows his dominant strategy, Owner 1’s payoff increases by $\alpha Q^{m} - C_{1}^{m}$ (by $2\alpha Q^{m} - C_{1}^{m}$) if she chooses I in the first period. Hence, choosing I is Owner 1’s dominant action if and only if $C_{1}^{m} < \alpha Q^{m}$ (if and only if $C_{1}^{m} < 2\alpha Q^{m}$); otherwise, NI is Owner 1’s dominant strategy.□

Proof of Result 3

The condition for investment in maintenance under an exogenously imposed redevelopment ($C_{i}^{m} < \alpha Q^{m}$; see Result 1) is more restrictive than the condition for investment in maintenance in the absence of a redevelopment option ($C_{i}^{m} < 2\alpha Q^{m}$; see Result 2).□

Proof of Result 4

The backward induction process, presented in “Appendix 2”, details the optimal choice at each node for any given set of model parameters. Specifically, Tables 1 and 2 detail the conditions under which Owner 1 and Owner 2, respectively, opt for investing in maintenance in the first period. Recall that following Result 1, the condition for owner i’s investment in maintenance when no-redevelopment is exogenously imposed is $C_{i}^{m} < 2\alpha Q^{m}$. As can be seen, the conditions listed in Tables 1 and 2 are never less, but are sometimes more, restrictive than the condition $C_{i}^{m} < 2\alpha Q^{m} .$ In other words, any owner who opts for no-investment in maintenance under an imposed no-redevelopment order, opts for no-investment in maintenance if redevelopment is endogenously determined. However, some owners who opt for investment in maintenance under an imposed no-redevelopment order opt for no-investment in maintenance if redevelopment is endogenously determined.□

Proof of Result 5

Let $Rent_{i}^{1}$ be the first-period rent collected by owner i. Owner i therefore opts for redevelopment (R) if and only if $Rent_{i}^{1} < Q^{h} - C_{i}^{h}$. Thus, the lower $Rent_{i}^{1}$, the more likely that owner i opts for R.□

Proof of Result 6

Consider the following conditions: (1) $Q^{h} - \alpha Q^{m} > C_{1}^{h} > Q^{h} - Q^{m}$; (2) $C_{2}^{h} < Q^{h} - Q^{m}$; (3) $C_{1}^{m} < \left( {2\alpha - 1} \right)Q^{m}$; (4) $C_{2}^{m} < \alpha Q^{m}$; and (5) $C_{2}^{m} < \left( {\alpha + 1} \right)Q^{m} - Q^{h} + C_{2}^{h}$. Under these conditions, according to the backward induction (see “Appendix 2”), Owner 2 at node <3> (node <2>) chooses between terminal nodes <17> and <18> (<15> and <13>), where $P_{2}^{17} > P_{2}^{18}$($P_{2}^{15} > P_{2}^{13}$).²¹ He therefore opts for NI at node <3> (opts for I at node <2>). Accordingly, Owner 1 at node <1> chooses between terminal nodes <17> and <15>, where $P_{2}^{17} > P_{2}^{15}$. She therefore opts for I at node <1>. Note that in this case $C_{2}^{m}$ is low enough for maintenance to be profitable for Owner 2, given exogenous implementation (or no implementation) of redevelopment. Yet by opting for NI, he reduces the rent of both units, and redevelopment thus becomes profitable for Owner 1.□

Proof of Result 7

Consider the following conditions: (1) $Q^{h} - \left( {1 - \alpha } \right)Q^{m} > C_{1}^{h} > Q^{h} - Q^{m}$; (2) $C_{2}^{h} < Q^{h} - Q^{m}$; (3) $C_{1}^{m} > \left( {2\alpha - 1} \right)Q^{m}$; (4) $C_{2}^{m} < \alpha Q^{m}$; and (5) $C_{2}^{m} > \left( {\alpha + 1} \right)Q^{m} - Q^{h} + C_{2}^{h}$. Under these conditions, according to the backward induction (see “Appendix 2”), Owner 2 at node <3> (node <2>) chooses between terminal nodes <17> and <18> (<15> and <13>), where $P_{2}^{17} > P_{2}^{18}$($P_{2}^{15} > P_{2}^{13}$). He therefore opts for NI at node <3> (I at node <2>). Owner 1 at node <1> accordingly chooses between terminal nodes <17> and <15>, where $P_{2}^{15} > P_{2}^{17}$. She therefore opts for NI at node <1>. Note that in this case $C_{1}^{m}$ is low enough for maintenance to be profitable for Owner 1, given the implementation of redevelopment. Yet by opting for NI, she assures Owner 2 that she will choose R, which in turn allows him to invest in maintenance.□

Proof of Result 8

Consider the following conditions: (1) both owners have identical parameters such that $C_{1}^{h} = C_{2}^{h} = C^{h}$ and $C_{1}^{m} = C_{2}^{m} = C^{m}$; (2) $C^{m} < \alpha Q^{m}$; (3) $p^{ml} = p^{m}$; (4) $\left( {p^{l} - p^{m} } \right) \left( {Q^{h} - C^{h} } \right) > \left( {1 + p^{m} } \right)Q^{m} - C^{m}$. Under these conditions, given the opportunity, both owners opt for redevelopment in the second period. Nevertheless, their likelihood to receive the opportunity decreases if any of them opt for investing in maintenance. According to the backward induction (see “Appendix 2”), Owner 2 at node <2> (node <3>) chooses between nodes <2A> and <2B> (<3A> and <3B>), where the payoff expectancy that follows investment is smaller (is larger) than the payoff expectancy that follows no-investment. He therefore opts for NI at node <2> (I at node <3>). Owner 1 at node <1> accordingly chooses between nodes <3B> and <2A>, where his payoff expectancy that follows investment is smaller than the payoff expectancy that follows no-investment. She therefore opts for NI at node <1>. Note that in this case $C_{1}^{m}$ and $C_{2}^{m}$ are low enough for maintenance to be profitable for both owners, given the implementation of redevelopment. Yet they both opt in equilibrium for NI, due to the associated increased likelihood for redevelopment.□

Appendix 2: A backward induction process

Owner 2’s optimal action at nodes <8> through <11>:

• Node <11>

  Owner 2 opts for R if $Q^{h} - C_{2}^{h} > Q^{m}$; otherwise he opts for NR

• Node <10>

  Owner 2 opts for R if $Q^{h} - C_{2}^{h} > \left( {1 - \alpha } \right)Q^{m}$; otherwise he opts for NR

• Node <9>

  Owner 2 opts for R if $Q^{h} - C_{2}^{h} > \alpha Q^{m}$; otherwise he opts for NR

• Node <8>

  Owner 2 opts for R if $Q^{h} - C_{2}^{h} > 0$; otherwise he opts for NR

Owner 1’s optimal action at nodes <4> through <7>:

• Node <7>

  Owner 1 opts for R if $Q^{h} - C_{1}^{h} > Q^{m}$; otherwise she opts for NR

• Node <6>

  Owner 1 opts for R if $Q^{h} - C_{1}^{h} > \alpha Q^{m}$; otherwise she opts for NR

• Node <5>

  Owner 1 opts for R if $Q^{h} - C_{1}^{h} > \left( {1 - \alpha } \right)Q^{m}$; otherwise she opts for NR

• Node <4>

  Owner 1 opts for R if $Q^{h} - C_{1}^{h} > 0$; otherwise she opts for NR

Owner 2’s optimal action at nodes <2> and <3>:

• Nodes <2> and <3>

  The backward induction process conducted above for the second period sub-game leads Owner 2 to essentially choose between 2 payoffs as he chooses I and NI at node <3> (node <2>). Table 1 below lists all possible outcome combinations and the conditions under which Owner 2 optimally opts for I. The shaded cells in the table indicate that no set of model parameters can lead Owner 2 to choose between these two alternative payoffs (given the analysis above); the values in gray correspond to the quality-dependent redevelopment incentives policy described in Sect. 4 and in Fig. 2. The table suggests that, given that the strategy of both owners in all the following nodes is R, Owner 2 opts for I at node <3> (<2>) only if $C_{2}^{m} < \alpha Q^{m}$; if the strategy of at least one owner is NR, regardless of his current choice, Owner 2 opts for I only if $C^{m} < 2\alpha Q^{m}$. Finally, otherwise (i.e., when implementation of redevelopment depends on his choice), Owner 2 opts for I only if $C_{2}^{m} < \left( {1 + \alpha } \right)Q^{m} - Q^{h} + C_{2}^{h}$ (only if $C_{2}^{m} < 2\alpha Q^{m} - Q^{h} + C_{2}^{h}$).

Table 1 Conditions for Owner 2 to Opt for I given the relevant terminal node couplet

Owner 1’s optimal action at node <1>:

• Node <1>

  The backward induction process above leads Owner 1 to essentially choose between two payoffs as she chooses I and NI at node <1>. Table 2 below lists all possible outcome combinations, and the conditions under which Owner 1 optimally opts for I. Shaded cells in the table indicate that no set of model parameters can lead Owner 1 to choose between these two alternative payoffs (given the analysis above). Note that all the conditions listed in Table 2 for Owner 1 to opt for I are at least as restrictive as $2Q^{m} > C_{1}^{m}$. In other words, when $2Q^{m} < C_{1}^{m}$, Owner 1’s optimal action at node <1> is NI. Results further suggest that there is no (non-negative) value of $C_{1}^{m}$ under which Owner 1 optimally opts for I without further conditions.

Table 2 Conditions for Owner 1 to Opt for I given the relevant terminal node couplet