Generalized Hyperbolic Discounting in Security Games of Timing
Abstract
:1. Introduction
2. Related Work
3. Discounting
 For $\alpha <\beta $ (superhyperbolic discounting), the area under $D\left(t\right)$’s curve is finite and given by:$$\begin{array}{c}\hfill {\int}_{\tau =0}^{+\infty}D\left(\tau \right)\mathrm{d}\tau =\frac{1}{\beta \alpha}.\end{array}$$
 For $\alpha \ge \beta $ (subhyperbolic discounting), ${\int}_{\tau =0}^{T}D\left(\tau \right)\mathrm{d}\tau $ does not converge for $T\to +\infty $.
4. Model
4.1. Overview
4.2. Player Strategies
4.3. Player Control
4.4. Player Utilities
4.4.1. Gains
4.4.2. Costs
4.5. Restricted Strategies
4.5.1. Exponential Strategies
4.5.2. Periodic Strategies with Random Phase
5. Analysis
5.1. SubHyperbolic Discounting
5.2. SuperHyperbolic Discounting
5.3. Player Utilities for SuperHyperbolic Discounting
5.3.1. For Exponential Play
5.3.2. For Periodic Play
5.4. Player Incentives and Best Responses
5.4.1. Directionality of Incentives and Base Incentive
 If her base incentive is negative, her best response is not to play.
 If her base incentive is strictly positive, then her incentive function has a single root at ${\nu}_{i}^{\u2605}>0$, and her best response is to play at rate ${\nu}_{i}^{\u2605}$.
 If her base incentive is strictly negative, then her unique best response is not to play.
 If her base incentive is zero, then any play rate ${\nu}_{i}\in [0,{\overline{\nu}}_{j}]$ is a best response.
 If her base incentive is strictly positive, then her incentive function has a single root at ${\nu}_{i}^{\u2605}>0$, and her best response is to play at rate ${\nu}_{i}^{\u2605}$.
5.4.2. Directionality and Origin of Base Incentive
 If $2{\alpha}_{A}\ge {\beta}_{A}$, move at nonzero rates, regardless of cost (${c}_{i}$).
 If $2{\alpha}_{A}<{\beta}_{A}$, move at nonzero rates iff ${c}_{A}<\frac{1}{{\beta}_{A}2{\alpha}_{A}}$ and at the zero rate otherwise.
5.5. Equilibria for SuperHyperbolic Discounting
5.5.1. NonParticipatory Equilibria
 If $2{\alpha}_{A}<{\beta}_{A}$ and ${c}_{A}\ge \frac{1}{2{\alpha}_{A}{\beta}_{A}}$, then there is a nonparticipatory equilibrium in which neither player moves.
 Otherwise, there may be an equilibrium in which only the attacker plays if $2{\alpha}_{D}<{\beta}_{D}$ or if $2{\alpha}_{D}={\beta}_{D}$ and ${c}_{D}\ge \frac{1}{{\alpha}_{D}}$.
Algorithm 1 Procedure for finding the set of nonparticipatory equilibria for exponential and periodic play 

5.5.2. Participatory Equilibria for Exponential Play
 ${\nu}_{D}^{\u2605}\in ]0,\phantom{\rule{0.166667em}{0ex}}{\overline{\nu}}_{D}[$, where ${\overline{\nu}}_{D}>0$ is the (unique) root of the attacker’s base incentive function.
 If $2{\alpha}_{D}\ge {\beta}_{D}$, then ${\nu}_{A}^{\u2605}\in ]0,\phantom{\rule{0.166667em}{0ex}}{\overline{\nu}}_{A}[$, where ${\overline{\nu}}_{A}>0$ is the root of the defender’s base incentive function.
 If $2{\alpha}_{D}<{\beta}_{D}$, then ${\nu}_{A}^{\u2605}\in ]{\overline{\nu}}_{A}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}{\overline{\nu}}_{A}^{\left(2\right)}[$, where ${\overline{\nu}}_{A}^{\left(1\right)}$ and ${\overline{\nu}}_{A}^{\left(2\right)}$ are the roots of the defender’s base incentive function with ${\overline{\nu}}_{A}^{\left(2\right)}\ge {\overline{\nu}}_{A}^{\left(1\right)}>0$.
5.5.3. Participatory Equilibria for Periodic Play
Algorithm 2 Algorithm for finding the set of participatory equilibria with fastermoving defender for periodic play 
// Check if the attacker’s base incentive has a root

6. Discussion
6.1. Degenerated Discounting Behavior
 All existing research and results on FlipItlike timing games without discounting carry over to the most commonly observed discounting behavior. This includes the results presented in van Dijk et al. [2] and followup work without discounting.
 The players’ utilities, incentives, best responses, and the game’s equilibria are not impacted by the fact that the defender starts the game in control of the resource. Differences between defender and attacker emerge only when discounting superhyperbolically.
Commitment
Organizational Actors
Rationalization of Risk
6.2. ShortHorizon and LongHorizon Discounting
6.3. Player Indifference for Periodic Play
6.4. Effective Move Cost and Motivation
 Increased effective cost. Players must pay for the instantaneous cost of a move “up front”. They first pay for the move and accrue benefits from this move later, resulting in the benefits being discounted at a higher rate than the costs. For both players, this effect results in a slightly higher “effective cost” of a move and reduces their incentive.
 Relevance of starting position. Another behavioral change is caused by the defender starting out in control of the resource. This envigorates the attacker—he will attack at reasonably high rates to reduce the time before the first move to obtain the resource before its value has been discounted away, even in the absence of a defender. In contrast, this demotivates the defender, as she is sure to be in control of the resource when it is at its most valuable.
6.5. Equilibrium Selection
 The attacker moves infrequently, and the defender does not respond.
 Both players move at a low, nonzero rate.
 Both players move at a higher, nonzero rate.
6.6. Perfectly Secure Systems
6.7. NonOptimality of Periodic Strategies
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Divergent Behavior
Appendix A.1. Gains
 ${s}_{k+1}={s}_{k}+{t}_{k}^{*}$
 ${t}_{k}^{*}\ge 1$
 ${lim}_{k\to \infty}{s}_{k}=+\infty $
 $\forall t\in {\mathbb{R}}^{+},\phantom{\rule{0.166667em}{0ex}}\exists !k,\phantom{\rule{0.166667em}{0ex}}{s}_{k}<t\le {s}_{k+1}$
 $\leftf\right({s}_{k},{t}_{k}^{*})v\le \epsilon $
 $\forall t>{t}_{k}^{*},\phantom{\rule{0.166667em}{0ex}}f({s}_{k},t)v<\epsilon $
 ${t}_{m}^{*}\ge c\xb7({s}_{m}+{s}_{m+1})$,
 $\leftf\right(0,{s}_{m})v<c\epsilon $,
 $\leftf\right({s}_{m},{t}_{m}^{*})v=\epsilon $.
Appendix A.2. Costs
Appendix B. Player Gains
Appendix B.1. Structural Aspects
Appendix B.2. Exponential Play
Appendix B.3. Periodic Play
Appendix C. Player Incentives
Appendix C.1. Expressions
Appendix C.1.1. Exponential Play
Appendix C.1.2. Periodic Play
Appendix C.2. Direction of Incentive
Appendix C.2.1. Exponential Play
 The exponential integral function ${E}_{r}\left(x\right)$ is positive:$$\begin{array}{c}\hfill {E}_{r3}\left(x\right)>0,\phantom{\rule{1.em}{0ex}}{E}_{r2}\left(x\right)>0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}_{r1}\left(x\right)>0.\end{array}$$
 The exponential integral function ${E}_{r}\left(x\right)$ is decreasing in its order r, that is, the partial derivative of ${E}_{r}$ to r is negative and$$\begin{array}{c}\hfill {E}_{r3}\left(x\right)>{E}_{r2}\left(x\right)>{E}_{r1}\left(x\right).\end{array}$$
 The rate at which the decrease happens is decreasing in the order r, that is, the secondorder partial derivative of ${E}_{r}$ to r is positive and consequently$$\begin{array}{c}\hfill {E}_{r3}\left(x\right){E}_{r2}\left(x\right)>{E}_{r2}\left(x\right){E}_{r1}\left(x\right).\end{array}$$
Appendix C.2.2. Periodic Play
 If she is the slowermoving player, her incentive is independent of her play rate.
 If she is the fastermoving player, her incentive is strictly decreasing in her play rate.
Appendix C.3. Base Incentives
 The attacker’s base incentive function is strictly decreasing.
 If $2{\alpha}_{D}\ge {\beta}_{D}$, then the defender’s base incentive function is strictly decreasing for strictly positive attack rates (${\nu}_{A}\in ]0,\phantom{\rule{0.166667em}{0ex}}+\infty [$).
 If $2{\alpha}_{D}<{\beta}_{D}$, then the defender’s base incentive function is first strictly increasing, then strictly decreasing.
Appendix C.4. Origin of Base Incentive
 It is equal to ${c}_{D}$ for ${\nu}_{A}=0$.
 If $2{\alpha}_{D}>{\beta}_{D}$, then it becomes unboundedly large for ${\nu}_{A}\downarrow 0$.
 If $2{\alpha}_{D}={\beta}_{D}$, then it is equal to $\frac{1}{{\alpha}_{D}}{c}_{D}$ for ${\nu}_{A}\downarrow 0$.
 If $2{\alpha}_{D}<{\beta}_{D}$, then it is equal to ${c}_{D}$ for ${\nu}_{A}\downarrow 0$.
 If $2{\alpha}_{A}\ge {\beta}_{A}$, then it becomes unboundedly large for ${\nu}_{D}\downarrow 0$.
 If $2{\alpha}_{A}<{\beta}_{A}$, then it is equal to $\frac{1}{{\beta}_{A}2\xb7{\alpha}_{A}}{c}_{A}$ for ${\nu}_{D}=0$.
Appendix D. A Renewal Strategy Beating the Periodic Strategy
Notes
1  
2  In Merlevede et al. [13], these parameters are named ${\lambda}_{D}$ and ${\lambda}_{A}$ instead of ${\beta}_{D}$ and ${\beta}_{A}$, and ${\mathsf{\Lambda}}_{D}$ and ${\mathsf{\Lambda}}_{A}$ instead of ${\beta}_{D}^{c}$ and ${\beta}_{A}^{c}$. 
3  A player can move at most once at a particular instance of time and a finite number of times over any finite time interval. 
4  We limit ourselves to cases where both players discount superhyperbolically and do not exhaustively cover scenarios where one player discounts superhyperbolically and another subhyperbolically. As there is no discontinuity in player best responses near $\alpha =\beta $, outcomes for mixed scenarios can be observed under the superhyperbolic discounting by bringing $\alpha $ and $\beta $ close to each other. 
5  This research does primarily focus on income or costs occurring at one particular point in time, not on continuous income and cost streams as is the case here. 
6  We pick a value for $r=\beta /\alpha $ and then compute parameters $\alpha $ and $\beta $ as $\alpha ={2}^{1/r}1$ and $\beta =\alpha \xb7r$. 
7  Define income resulting from a move as follows: (•) When a player in control of the resource performs a move, it does not result in income. (•) When a player not in control of the resource performs a move, flipping the resource, it results in income equal to the value generated by the resource from the time of the move until the resource flips again. Start by splitting the game into two parts: the part of the game before the attacker’s first move (ante) and the part after the attacker’s first move (post). • (ante) If the attacker is the slower player, this part of the game always yields him precisely zero gain, irrespective of either player’s play rate. If the defender is the slower player, this part of the game does yield her gain, but the expected amount only depends on the duration of ante, which is also independent of her play rate. • (post) The probability density of the slower player flipping at any time is constant and equal to her play rate. Every one of her moves results in an expected income; while difficult to determine this income exactly, it is independent of her own play rate, as it is certain to result in a change of ownership while the value of ownership depends only on the time and the play rate of the faster player. The slower player’s gain is, therefore, proportional to her play rate, and her incentive is independent of her play rate. 
8  To be precise, the periodic strategies strictly dominate the class of nonarithmetic or nonlattice renewal strategies. Arithmetic strategies are strategies for which all possible realizations happen at an integer multiple of some real number. 
9  The CDF for the time of the first move is $1(1{t}_{0}{\nu}_{f}){\overrightarrow{I}}_{{t}_{0}\le 1/{\nu}_{f}}\xb7(1t{\nu}_{s}){\overrightarrow{I}}_{{t}_{0}\le 1/{\nu}_{s}}={t}_{0}({\nu}_{f}+{\nu}_{s}{t}_{0}{\nu}_{f}{\nu}_{s}){\overrightarrow{I}}_{{t}_{0}\le 1/{\nu}_{f}}$, where $\overrightarrow{I}$ is the indicator function. The PDF for the time of the first move is its derivative, $p\left({t}_{0}\right)=j({\nu}_{f}+{\nu}_{s}2t+0{\nu}_{f}{\nu}_{s}){\overrightarrow{I}}_{{t}_{0}\le 1/{\nu}_{f}}$. An expression for the total anonymous gain is, therefore,
$$\begin{array}{c}\hfill (\beta \alpha ){\int}_{{t}_{0}=0}^{+\infty}p\left({t}_{0}\right){\int}_{\tau ={t}_{0}}^{+\infty}{D}_{i}\left(\tau \right)\mathrm{d}\tau \mathrm{d}{t}_{0}={\int}_{{t}_{0}=0}^{1/{\nu}_{f}}\frac{{\nu}_{f}+{\nu}_{s}2{t}_{0}{\nu}_{f}{\nu}_{s}}{{(1+{t}_{0}\alpha )}^{\frac{\beta \alpha}{\alpha}}}if{t}_{0}.\end{array}$$
We can confirm that this expression is equal to the sum of ${\overline{G}}_{s}$ and ${\overline{G}}_{f}$ as presented in Lemmas A16 and A17. 
10  The probability of the faster player having moved last at any point in time is equal to ${\int}_{\tau =0}^{1/{\nu}_{f}}{\nu}_{f}\xb7\tau \xb7{\nu}_{s}\mathrm{d}\tau =\frac{{\nu}_{s}}{2{\nu}_{f}}$. The slower player moved last with probability $1\frac{{\nu}_{s}}{2{\nu}_{f}}$. 
11  The expected duration of the intervals owned by the fastermoving player is ${\int}_{\tau =0}^{1/{\nu}_{f}}(1\frac{\tau}{{\nu}_{s}})\mathrm{d}\tau =\frac{1}{{\nu}_{f}}\frac{{\nu}_{s}}{2{\nu}_{f}^{2}}$. The intervals of the slowermoving player have a shorter expected length of ${\int}_{\tau =0}^{1/{\nu}_{f}}(1\frac{\tau}{{\nu}_{f}})\mathrm{d}\tau =\frac{1}{2{\nu}_{f}}$. 
12  https://dlmf.nist.gov/8.19.E15, (accessed on 10 November 2023). 
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Class of Discounting  No Discounting [2]  Exponential Discounting [14]  Generalized Hyperbolic Discounting 

Discounting parameters  ∅  $\{{\beta}_{A},{\beta}_{A}^{c},{\beta}_{D},{\beta}_{D}^{c}\}$2  $\{{\alpha}_{D},{\beta}_{D},{\alpha}_{D}^{c},{\beta}_{D}^{c},{\alpha}_{A},{\beta}_{A},{\alpha}_{A}^{c},{\beta}_{A}^{c}\}$ 
Relation to other models  n.a.  $lim\lambda \to 0$ is None  $lim\alpha \to 0$ is Exponential 
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Merlevede, J.; Johnson, B.; Grossklags, J.; Holvoet, T. Generalized Hyperbolic Discounting in Security Games of Timing. Games 2023, 14, 74. https://doi.org/10.3390/g14060074
Merlevede J, Johnson B, Grossklags J, Holvoet T. Generalized Hyperbolic Discounting in Security Games of Timing. Games. 2023; 14(6):74. https://doi.org/10.3390/g14060074
Chicago/Turabian StyleMerlevede, Jonathan, Benjamin Johnson, Jens Grossklags, and Tom Holvoet. 2023. "Generalized Hyperbolic Discounting in Security Games of Timing" Games 14, no. 6: 74. https://doi.org/10.3390/g14060074