Intergenerational game theory
Bargaining with the unborn
2026-05-04 — 2026-05-04
In Which the Ramsey Growth Problem Is Salvaged From Its Representative-Agent Simplicity, the Overlapping Generations Model Is Quoted to Map Intergenerational Theft, and the Future Is Detained by the Present.
Content warning:
AI Slop awaiting human comprehension and review.
Longtermists wonder what we should give the people of the remote future because of their intrinsic worth. This notebook is not about that. This is about what the future can extract from the present, with what little bargaining leverage it can muster.
Most game theory assumes the players exist at the same time and can interact: observe each other’s moves, retaliate against defections, form coalitions, bargain over outcomes. Intergenerational games break all of this, in favour of the actually-existing.
Future generations cannot retaliate against past ones; they do not exist when the moves are made in the intergenerational game. They cannot voluntarily enter coalitions; they have no preferences yet to voice. They cannot reject offers; the bargain arrives as a fait accompli.
As such, the standard equilibrium prediction of self-interested rational play promises nothing good for the unborn. Indeed, it delivers expropriation: environmental depletion, sovereign debt, exhausted commons, accumulated tail risk. These are not coordination failures between present-day parties; they are first-mover advantages against people who cannot yet object (Kotlikoff and Rosenthal 1993).
This is the inter-personal counterpart of intertemporal decisions, where I treat the intra-personal version (Strotz 1955; Pollak 1968; McClennen 1990). This seems to me to have important structural commonalities — present moves bind future agents who could not participate — but here the future agents are entirely different people, not longitudinal selves in the same individual. Maybe these cash out into the same problem under multi-level agency?
1 Oh damn, iterated game theory does not save us
There is a stylized insight in iterated games that tells us that populations can attain a prosperous tit-for-tat equilibrium, i.e. mutual cooperation sustained by the threat of retaliation in repeated play. This depends on retaliation being available and credible, which it is not, against the arrow of time. The shadow of the future, in Axelrod’s phrase, has to fall on a player who exists. When the players never meet, the shadow falls on no one.
The same goes for the bargaining-set / class-emergence dynamics from Axtell, Epstein, and Young (2001)’s multi-agent bargaining model: those require iterated negotiation between the two parties to the bargain. With future generations, only one party is present in any given negotiation.
2 Overlapping generations
The standard formal device for getting some purchase on the problem is the Overlapping Generations (OLG) model (Samuelson 1958; Diamond 1965). The basic insight is that generations are not point-events but cohorts with finite lifespans, and at any moment several cohorts coexist. A grandparent, parent, and child can interact directly. They have overlapping interests with neighbouring cohorts, who have overlapping interests with theirs in turn. This gives a transitive chain of reciprocal interactions linking the present to the distant future, without requiring any single agent to bargain with someone unborn.
OLG is enough to recover some intergenerational cooperation under the right preference structures — altruism toward children, social-norm enforcement, public-pension institutions, family-firm continuity. But it does not give a fairness theorem for distant generations. The cooperation chain attenuates with each link: future-future-future agents are weighted only as much as their immediate ancestors cared about their children’s children’s children, which is generally not very much.
3 Confucius saw this one coming
Confucianism seems to have a strong emphasis on filial piety, which prescribes respect and care for one’s parents and ancestors, and by extension, a moral framework for considering the welfare of future generations.
Does it work? Is the problem here a me problem, and appropriately confucian successors would be able to solve this better than filially impious li’l me?
🚧TODO🚧 From here onwards this notebook is slop research notes for a break out session that you were probably not a participant in. Stop reading if you value your eyeballs.
4 Rescuing games between cohorts
Phelps and Pollak (1968) turn the OLG setup into an explicit non-cooperative game with one move per cohort. A cohort inherits a stock of capital, chooses how much of it to consume and how much to leave behind, and then exits. Its payoff is its own consumption plus some weight on the welfare of its descendants — a weight strictly less than one, since concern for people we will never meet is imperfect. What makes this a game rather than a plan is that a cohort cannot dictate what its heirs do with the bequest. The heirs face the same problem in turn, against their heirs.
So the solution concept is subgame-perfect equilibrium: every cohort’s saving rate is a best response to the saving rates its successors will in fact choose, given that those are themselves best responses further down the line. The resulting path is generally not the one a planner would pick if it could dictate the whole sequence of saving rates at once, and it can miss by a lot. No cohort does anything irrational, each responds correctly to the situation it is handed, and the sequence still comes out worse for everybody than a path all of them would have preferred.
Formally this is the same object as one person’s saving plan under non-exponential discounting (Strotz 1955; Pollak 1968; McClennen 1990): substitute “the self at time \(t\)” for “cohort \(t\)” and the equations do not change. The history runs the opposite way to how it is usually told, though. The \(\beta\)–\(\delta\) quasi-hyperbolic discount function that Laibson later made a staple of behavioural economics is Phelps and Pollak’s, and it started life as a model of generations, not of selves.
McClennen’s taxonomy of naïve, sophisticated and resolute choosers carries over to cohorts, two of the three intact.
A naïve cohort saves as though its descendants will stick to the plan it has in mind for them. They will not, because they inherit the same temptation and re-optimize when their turn comes. Unlike the naïve self, who is repeatedly surprised, the naïve cohort never finds out; it is dead before the evidence arrives.
A sophisticated cohort works out what its successors will actually do and best-responds to that, which is the subgame-perfect path above. Sophistication buys accuracy rather than a better outcome. Correctly anticipating that our heirs will consume most of what we leave them is a reason to leave less, not more.
A resolute chooser, in McClennen’s sense, adopts a plan and then carries it out, treating its successive selves as one agent rather than as opponents to be out-guessed; no external commitment device is needed, because the resoluteness is internal. That is the one with nowhere to live here. Resoluteness requires an agent persisting from the moment a plan is adopted to the moment it must be honoured. Generational turnover is the absence of any such agent. The cohort that adopts the plan is gone before it comes due, and the cohort that has to honour it never agreed to anything.
Intergenerational resoluteness therefore has to be manufactured, by stipulation or by institution — a constitution, a sovereign wealth fund, a central-bank mandate, a treaty with a long ratchet — each of which is an attempt to build a super-agent that outlives every cohort taking part in it. This is the same move as the commitment devices of the intra-personal case, except that here it is not optional, there being no continuous agent to fall back on. It relocates the problem rather than dissolving it: whether the manufactured super-agent stays resolute is a question about its incentives, which returns us to multi-level agency.
Which mathematical object we land in depends on the shape of the altruism. Non-exponential weighting of descendants — the \(\beta\)–\(\delta\) form, a discrete drop between us and them followed by gentler discounting thereafter — turns the Ramsey growth problem from a single optimization into a Markov game whose players are cohorts. Exponential weighting does not. Under the dynastic altruism of Barro (1974), where a parent weights the child’s whole utility by a constant factor and the child does the same for the grandchild, the weights compose exponentially down the line; every cohort then ranks future consumption streams identically, and the game collapses back to a single representative-agent planner solving the ordinary Ramsey problem. That collapse is the formal version of “don’t die” below. A dynasty that discounts consistently is, as far as the model can tell, one immortal agent.
5 Axiomatic intergenerational equity
Most of the formal work on intergenerational fairness is not game theory; it is axiomatic social choice over infinite streams of well-being. Each generation gets a well-being level \(x_t\), and the question is which orderings of infinite streams \((x_1, x_2, \dots)\) are consistent with two minimal-seeming axioms:
- Strong Pareto (sensitivity): if one generation is strictly better off and none worse, prefer the new stream.
- Finite Anonymity (equal treatment): permuting finitely many generations leaves the ranking unchanged.
There are various impossibility results — Basu–Mitra (Basu and Mitra 2003), Lauwers, Zame, and others — blocking us from jointly satisfying these axioms (Asheim 2010). There is no numerically representable social welfare function that satisfies both (Diamond–Basu–Mitra), and if we drop numerical representability and just ask for an explicitly definable complete ordering, that fails too (Lauwers–Zame). One of the axioms has to weaken or go.
The literature has taken two paths through this.
Keep both axioms, give up completeness. Incomplete orderings like utilitarian or leximin “overtaking” criteria can compare some pairs of infinite streams but not all. Some intergenerational dilemmas come out unrankable, which is awkward for policy advice but at least flagged rather than fudged.
Keep completeness, give up equal treatment. Discounted utilitarianism — the standard policy-economics workhorse — applies a discount factor \(\delta < 1\) each period, weighting future generations less. Chichilnisky (1996) names this a Dictatorship of the Present, since beyond some finite time the future stops mattering for the ranking. The Stern–Nordhaus debate over climate policy is largely a debate over how much of this dictatorship to accept.
A third path tries to weaken the axioms in the right direction rather than throwing one out. Asheim’s Hammond Equity for the Future (HEF) is one such weakening. Standard Hammond equity says any transfer from a better-off generation to a worse-off one improves the social ranking, provided the ordering is preserved. HEF restricts this to one specific case — when the present is better off than an infinite number of future generations, and a present sacrifice would benefit all of them while keeping the present better off than the future. My understanding of the intuition here is that a small present sacrifice is worth it if it benefits an infinite number of unambiguously-worse-off future cohorts, while we should not demand maximal sacrifice from a barely-worse-off present to benefit a barely-worse-off future. Asheim, Mitra, and Tungodden show that HEF combined with weakened Pareto and continuity axioms gives a class of sustainable recursive social welfare functions, orderings that allow present sacrifice when the future is worse off but not when the future is already better off (Asheim, Mitra, and Tungodden 2012).
In productive technological environments — Asheim, Buchholz, and Tungodden’s “immediate productivity” condition — Strong Pareto and Finite Anonymity together imply that only non-decreasing well-being streams are undominated (Asheim, Buchholz, and Tungodden 2001). So sustainability is what the minimal-equity axioms imply when the technology can in fact produce things, not an additional normative add-on but a consequence of equality + sensitivity once productivity is in the picture.
The methodological frame across all of this is Rawls’s reflective equilibrium (Rawls 1971). Criteria for intergenerational equity should be judged on the ethical axioms they build on, and also on their consequences in concrete technological environments. If a criterion says we should sacrifice everything for any-positive-gain-by-the-future, or never sacrifice anything no matter how much the future gains, our intuition that both are wrong is itself evidence to revise the axioms. Picking axioms in the abstract and applying them blindly is bad methodology; we iterate between axioms and downstream implications until both feel right.
I started with Asheim’s Annual Review survey (Asheim 2010).
6 Veto-power formalism
A bargaining-flavoured strand of the axiomatic literature models the game as if future generations could refuse. Suppose each generation has a Rubinstein-style alternating-offers position (Rubinstein 1982) with a credible veto over others’ resource decisions (Manzini 2010). Even when future cohorts cannot in fact participate, asking “what allocation would they accept under bargaining parity?” gives a well-defined Social Welfare Relation that excludes the worst expropriations.
Such a veto is more a normative device than a positive one. It tells us what fairness would require if the unborn could speak; it does not predict what self-interested play will produce.
7 Proxy mechanisms in practice
Closer to political economy: institutions that artificially weight the bargaining power of those representing future interests.
Youth quotas in legislative bodies. Younger voters and legislators have longer remaining horizons and internalise more of the long-term consequences; their interests partially overlap with the unborn. The argument is roughly: youths are the next-best proxy for those who cannot yet vote — see the 2015 Intergenerational Justice Review issue for several treatments.
Future-generations tribunals or commissioners, with explicit review or veto powers over policies that bind the long term. Hungary, Wales, Israel, and others have variants; their bite varies enormously by jurisdiction and political will. See Basson et al. (2025) on the human-rights framing.
Constitutional limits on the actions of a present majority — debt ceilings, environmental amendments, supermajority requirements for long-term commitments. The constitutional framing serves as a proxy commitment device — see commitment — that binds future selves of the polity.
These are all imperfect. Youths are not the unborn; commissioners have their own interests; constitutions can be amended. But they are practical institutions trying to do the work the formal models say is needed.
8 Discounting problems
A deep open question is how I should rationally discount future utility when other people will experience it than me, because I will be dead. A positive discount rate — the empirical norm in cost-benefit analysis — heavily reduces the present-day weight on far-future welfare; this is what allows current generations to rationalise expropriation in policy documents. A zero or near-zero rate produces unbounded numbers (any benefit that compounds forever has infinite present value), forcing modellers to introduce extinction probabilities or other regularizations whose values are themselves contestable.
One area where people have though long and hard about this is climate change. The Stern–Nordhaus debate on climate-change policy is largely a debate about discounting: the formal cost-benefit models produce wildly different recommendations depending on a parameter that game theory itself does not pin down. There is no equilibrium-style answer. Famous treatments such as DeCanio’s Game Theory and Climate Change (DeCanio and Fremstad 2013) frame it as multi-objective optimization under deep uncertainty, with the discount rate as a normative input rather than a derivable feature.
9 Don’t die
One way to ameliorate intergenerational games is to not have any generations. If we can achieve indefinite lifespans, then the future is no longer a different generation; it’s just a later self. OTOH, humans can also suck at helping out their future selves.
10 Connections
Adjacent to: cooperation (mechanisms generally), coalition games (when “the future generation” is itself heterogeneous), commitment (intertemporal commitment as the dual problem), collective action (climate and commons specifically), game complexity (computing fair allocations in OLG with many cohorts is its own complexity story), and economics of growth (the standard growth models — Solow, Ramsey, Dasgupta–Heal–Solow — are doing intergenerational welfare analysis under a particular axiom set, usually discounted utilitarianism, often without flagging the choice).
Longtermism (the EA / x-risk strand) sits inside this axiomatic space — typically a near-zero discount rate plus expected-total-utility aggregation, which is one specific resolution of the SP + FA tension. Whether longtermists engage with the impossibility results explicitly, or just smuggle in their preferred resolution via expected utility, varies a lot.`
AI successionism — that AI systems should or will succeed humanity, and that this is acceptable or even desirable — fits inside the same frame. The problem is sharpened, though, because the “future generations” might have radically different utility functions and need not be biologically continuous with us at all. I am noodling on this elsewhere; the axiomatic frame is one tool for asking which axioms successionism implicitly endorses or rejects.
11 Incoming
Samuel Moyn, The dictatorship of the elderly
Louise Perry Modernity’s Self-Destruct Button
The Ponzi scheme of the old age pension is already collapsing, and it will almost certainly have collapsed entirely by the time I reach my sixty-eighth birthday. A gerontocratic voter base will cling to the state pension as long as possible—I wouldn’t be surprised if the British government abolished universal education before it abolished the state pension.